summaryrefslogtreecommitdiff
path: root/macros/latex/contrib/polynom/polynom.dtx
blob: c85ba214fa89cad05a06013c434ea0414f5524a0 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
3668
3669
3670
3671
3672
3673
3674
3675
3676
3677
3678
3679
3680
3681
3682
3683
3684
3685
3686
3687
3688
3689
3690
3691
3692
3693
3694
3695
3696
3697
3698
3699
3700
3701
3702
3703
3704
3705
3706
3707
3708
3709
3710
3711
3712
3713
3714
3715
3716
3717
3718
3719
3720
3721
3722
3723
3724
3725
3726
3727
3728
3729
3730
3731
3732
3733
3734
3735
3736
3737
3738
3739
3740
3741
3742
3743
3744
3745
3746
3747
3748
3749
3750
3751
3752
3753
3754
3755
3756
3757
3758
3759
3760
3761
3762
3763
3764
3765
3766
3767
3768
3769
3770
3771
3772
3773
3774
3775
3776
3777
3778
3779
3780
3781
3782
3783
3784
3785
3786
3787
3788
3789
3790
3791
3792
% \iffalse
%
% The files  polynom.dtx  and  polynom.ins  and all files generated
% from these two files are referred to as `this work'.
%
% This work is copyright 2000-2017 Carsten Heinz, Hendri Adriaens.
%
% This work may be distributed and/or modified under the conditions
% of the LaTeX Project Public License, either version 1.3 of this
% license or (at your option) any later version.
% The latest version of this license is in
%   http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2003/12/01 or later.
%
% This work has the LPPL maintenance status "maintained".
%
% The Current Maintainer of this work is Hendri Adriaens.
%
%<*driver>
\documentclass{ltxdoc}
\usepackage{hyperref,polynom}

\DisableCrossrefs
\OnlyDescription

\begin{document}
    \DocInput{polynom.dtx}
\end{document}
%</driver>
% \fi
%
%^^A
%^^A  Some definitions used for documentation.
%^^A
% \let\packagename\textsf
% \newenvironment{describe}{\trivlist\item[]}{\endtrivlist}
% \makeatletter
% \def\fps@figure{htbp}
% \let\c@table\c@figure
% \let\fps@table\fps@figure
% \makeatother
%^^A
%^^A  end of these definitions
%^^A
%
%\newbox\abstractbox
%\setbox\abstractbox=\vbox{
%   \begin{abstract}
%   The \packagename{polynom} package implements macros for manipulating
%   polynomials. For example, it can typeset polynomial long divisions and
%   synthetic divisions (Horner's scheme), which can be shown step by step.
%   The main test case and application is the polynomial ring in one variable
%   with rational coefficients.
%   \emph{Please note that this is work in progress. Multivariate polynomials
%   are \emph{currently} not supported.}
%   \end{abstract}}
%
% \title{The \packagename{Polynom} Package}
% \author{Copyright 2000--2017\\ Carsten Heinz \texttt{<\,cheinz@gmx.de\,>}, Hendri Adriaens}
% \date{2017/07/17\enspace Version 0.19\\ \box\abstractbox}
% \maketitle
% \section{Preface}
%
%Because Carsten Heinz could not be reached anymore for a long time,
%this package has been taken over according to the LPPL for
%maintenance by Hendri Adriaens. This package was using and
%redefining internals of the \packagename{keyval} package and hence
%it was incompatible with \packagename{xkeyval}. This problem has
%been solved and the processing of the \texttt{vars} key has been
%simplified. All following text is the original by Carsten Heinz.
%\hfill\emph{Hendri Adriaens, 2006/04/20}\\
%We thank Walter Daems for providing the \texttt D style.\hfill\emph{2016/12/09}\\
%And Hendrik Vogt for providing a bug fix on \texttt{\textbackslash polylongdiv}.\hfill\emph{2017/07/17} 
%
% \section{Introduction}
%
% Donald Arseneau has contributed a lot of packages to the \TeX\ community.
% In particular, he posted macros for long division on \texttt{comp.text.tex},
% which were also published in the TUGboat \cite{TUGboat} and eventually as
% \texttt{longdiv.tex} on CTAN. The \packagename{polynom} package allows to do
% the job with polynomials, see figure~\ref{division}. There you can also
% see an example of Horner's scheme for synthetic division.
% \begin{figure}
% \centering
% \begin{minipage}{.42\linewidth}
%   \[\polylongdiv{(X-1)(X^2+2X+2)+1}{X-1}\]
% \end{minipage}
% \hfil
% \begin{minipage}{.5\linewidth}
%   \[\polyhornerscheme[x=1]{x^3+x^2-1}\]
% \end{minipage}
%
% \begin{minipage}{.42\linewidth}
% \centering |\polylongdiv{X^3+X^2-1}{X-1}|
% \end{minipage}
% \hfil
% \begin{minipage}{.5\linewidth}
% \centering |\polyhornerscheme[x=1]{x^3+x^2-1}|
% \end{minipage}
% \caption{Polynomial long division and synthetic division. The commands both
%  are able to generate partial output, see \href{polydemo.pdf}{polydemo.pdf}
%  in fullscreen mode.}
% \label{division}
% \end{figure}
%
% \begin{figure}
%   \[\polylonggcd {(X-1)(X-1)(X^2+1)} {(X-1)(X+1)(X+1)}\]
% \centering |\polylonggcd {(X-1)(X-1)(X^2+1)} {(X-1)(X+1)(X+1)}|
% \caption{Euclidean algorithm with polynomials; the last nonzero remainder
%          is a greatest common divisor. In the case here, it is uniquely
%          determined up to a scalar factor, so \(X-1\) and \(\frac49X-\frac49\)
%          are both greatest common divisors}\label{euclidean}
% \end{figure}
%
% \begin{figure}
% \centering
% \begin{tabular}{ll}
% |\polyfactorize {(X-1)(X-1)(X^2+1)}|&\polyfactorize{(X-1)(X-1)(X^2+1)}\\ \\
% |\polyfactorize {2X^3+X^2-7X+3}|\\
% \multicolumn{2}{l}{\hspace*{.3\linewidth}\polyfactorize{2X^3+X^2-7X+3}}\\ \\
% \multicolumn{2}{l}{\makeatletter\ttfamily
%    \def\temp{\polyfactorize{120X^5-274X^4+225X^3-85X^2+15X-1}}^^A
%    \csname strip@prefix\expandafter\endcsname\meaning\temp}\\
% \multicolumn{2}{l}{\hspace*{.3\linewidth}^^A
%                    \polyfactorize{120X^5-274X^4+225X^3-85X^2+15X-1}}\\
% \end{tabular}
% \caption{Factorizations of some polynomials}\label{factorize}
% \end{figure}
%
% Figures~\ref{euclidean} and \ref{factorize} show applications of polynomial
% division. On the one hand the Euclidean algorithm to determine a greatest
% common divisor of two polynomials, and on the other the factorization of
% a polynomial with at most two nonrational zeros. This should suffice for many
% teaching aids.
%
%
% \section{Hints}
%
% As the examples show, the commands get their data through mandatory and
% optional arguments. Polynomials are entered as you would type them in math
% mode:\footnote{The scanner is based on the scanner of the \texttt{calc}
% package \cite{calc}. Read its documentation and the implementation part here
% if you want to know more.} you may use |+|, |-|, |*|, |\cdot|, |/|, |\frac|,
% |(|, |)|, natural numbers, symbols like |e|, |\pi|, |\chi|, |\lambda|, and
% variables; the power operator |^| with integer exponents can be used on
% symbols, variables, and parenthesized expressions.
% Never use variables in a nominator, denominator or divisor.
%
% The support of symbols is very limited and there is neither support of
% functions like \(\sin(x)\) or \(\exp(x)\), nor of roots or exponents other
% than integers, for example \(\sqrt\pi\) or \(e^x\). For teaching purposes
% this shouldn't be a major drawback. Particularly because there is a simple
% workaround in some cases: the package doesn't look at symbols closely,
% so define a function like \(e^x\) or `composed symbol' like \(\sqrt\pi\)
% as a symbol. Take a look at figure~\ref{epowerx} for an example.
% \begin{figure}
% \newcommand\epowerx{e^x}
% \[\polylongdiv[style=C,div=/]{\epowerx x^3-\epowerx x^2+\epowerx x-\epowerx}{x-1}\]
% \begin{verbatim}
%    \newcommand\epowerx{e^x}
%    \[\polylongdiv{\epowerx x^3-\epowerx x^2+\epowerx x-\epowerx}{x-1}\]\end{verbatim}
% \caption{Avoiding problems with \(e^x\). Be particularly careful in such
% cases. \emph{You} have to take care of the correct result \emph{since} the
% package does the computation. And by the way, it's always good to keep an
% eye on plausibility of the results}
% \label{epowerx}
% \end{figure}
%
% \medbreak
%
% Optional arguments are used to specify more general options (and also for
% the evaluation point for Horner's scheme). The options are entered in
% key=value fashion using the \packagename{keyval} package \cite{keyval}.
% The available options are listed in the respective sections below.
%
%
% \section{Commands}
%
%
% \subsection[\texttt{\textbackslash polyset}]
% {\normalfont\texttt{\textbackslash polyset}\marg{key=value list}}
%
% Keys and values in optional arguments affect only that particular operation.
% |\polyset| changes the settings for the rest of the current environment or
% group. This could be a single figure or the whole document. Almost every key
% described in this manual is allowed\,---\,just try it and you'll see.
% Table~\ref{keys} lists all keys, which are not connected to a particular
% command. An example is
% \begin{verbatim}
%    \polyset{vars=XYZ\xi,  % make X, Y, Z, and \xi into variables
%             delims={[}{]}}% nongrowing brackets\end{verbatim}
% Note that is essential to use \texttt{vars}-declared variables only.
% The package can't guess your intention and
% |\polylongdiv{\zeta^3+\zeta^2-1}{\zeta-1}|
% would divide a constant by a constant without the information $\zeta$ being
% a variable.
%
% \begin{table}
% \centering
% \begin{tabular}{p{.4\textwidth}p{.5\textwidth}}
% \texttt{vars=}\meta{token string}
%       & make each token a variable\\
%       &\\
% \texttt{delims=}\marg{left}\marg{right}
%       & define delimiters used for printing\\
%       & parenthesized expressions\\
% \end{tabular}
% \caption{General keys. Default for \texttt{vars} is \texttt{Xx}. The key
%          \texttt{delims} has in fact an optional argument which takes
%          invisible versions of the left and right delimiter. The default is
%          \texttt{delims=[\{\textbackslash left.\}\{\textbackslash right.\}]\{\textbackslash left(\}\{\textbackslash right)\}}
%          }\label{keys}
% \end{table}
%
%
% \subsection[\texttt{\textbackslash polylongdiv}]
% {\normalfont\texttt{\textbackslash polylongdiv}\oarg{key=value list}\meta{polynomial \(a\)}\meta{polynomial \(b\)}}
%
% The command prints the polynomial long division of $a/b$. Applicable keys
% are listed in table~\ref{keys:longdiv}. Of course, \texttt{vars} and
% \texttt{delims} can be used, too.
%
% \begin{table}
% \centering
% \begin{tabular}{p{.4\textwidth}p{.5\textwidth}}
% \texttt{stage=}\meta{number}
%       & print long division up to stage \meta{number} (starting with 1)\\
%       &\\
% \texttt{style=}\texttt{A$\vert$B$\vert$C$\vert$D}
%       & define output scheme for long division, refer \href{polydemo.pdf}{polydemo.pdf}\\
%       &\\
% \texttt{div=}\meta{token}
%       & define division sign for \texttt{style=C}, default is $\div$\\
% \end{tabular}
% \caption{Keys and values for polynomial long division. \texttt{style=A}
%          requires \texttt{stage=}\({}3\times(\#\)quotient's summands\()+1\)
%          to be carried out fully. The other styles \texttt{B} and \texttt{C}
%          need one more stage if the remainder is nonzero}
% \label{keys:longdiv}
% \end{table}
%
%
% \subsection[\texttt{\textbackslash polyhornerscheme}]
% {\normalfont\texttt{\textbackslash polyhornerscheme}\oarg{key=value list}\meta{polynomial}}
%
% The command prints Horner's scheme for the given polynomial with respect to
% the specified evaluation point. Note that the latter one is entered as a
% key=value pair in the form \meta{variable}\texttt{=}\meta{value}.
% Table~\ref{keys:horner} lists other keys and their respective values.
%
% \begin{table}
% \centering
% \begin{tabular}{p{.4\textwidth}p{.5\textwidth}}
% \meta{variable}\texttt{=}\meta{value}
%       & The definition of the evaluation point is \emph{mandatory}!\\
%       &\\
% \texttt{stage=}\meta{number}
%       & print Horner's scheme up to stage \meta{number} (starting with 1)\\
%       &\\
% \texttt{tutor=}\texttt{true$\vert$false}
%       &turn on and off tutorial comments\\
% \texttt{tutorlimit=}\meta{number}
%       & illustrate the recent \meta{number} steps\\
% \texttt{tutorstyle=}\meta{font selection}
%       & define appearance of tutorial comments\\
%       &\\
% \texttt{resultstyle=}\meta{font selection}
%       & define appearance of the result\\
% \texttt{resultleftrule=}\texttt{true$\vert$false}\newline
% \texttt{resultrightrule=}\texttt{true$\vert$false}\newline
% \texttt{resultbottomrule=}\texttt{true$\vert$false}
%       & control rules left to, right to, and at the bottom of the result\\
%       &\\
% \texttt{showbase=}\texttt{false$\vert$}\newline\phantom{\texttt{showbase=}}\texttt{top$\vert$middle$\vert$bottom}
%       & define whether and in which row the base (the value) is printed\\
% \texttt{showvar=}\texttt{true$\vert$false}
%       & print or suppress the variable name (additionally to the base)\\
% \texttt{showbasesep=}\texttt{true$\vert$false}
%       & print or suppress the vertical rule\\
%       &\\
% \texttt{equalcolwidth=}\texttt{true$\vert$false}
%       & use the same width for all columns or use their individual widths\\
% \texttt{arraycolsep=}\meta{dimension}
%       & space between columns\\
% \texttt{arrayrowsep=}\meta{dimension}
%       & space between rows\\
%       &\\
% \texttt{showmiddlerow=}\texttt{true$\vert$false}
%       & print or suppress the middle row\\
% \end{tabular}
% \caption{Keys and values for Horner's scheme. Don't use \texttt{showmiddlerow=false}
%          with \texttt{tutor=true}.}
% \label{keys:horner}
% \end{table}
%
% \iffalse
% The following key are not listed above:
%
%           mul=<math tokens>                           \cdot
%           plusface=left|right                         right
%           plusyoffset=<dimension>                     0pt
%
%           downarrow=<picture tokens>                  {\vector(0,-1){2.5}}
%           diagarrow=<picture tokens>                  {\vector(2,1){1.6}}
%           downarrowxoffset=<dimension>                0pt
%           diagarrowxoffset=<dimension>                0pt
% \fi
%
%
% \subsection[\texttt{\textbackslash polylonggcd}]
% {\normalfont\texttt{\textbackslash polylonggcd}\oarg{key=value list}\meta{polynomial \(a\)}\meta{polynomial \(b\)}}
%
% The command prints equations of the Euclidean algorithm used to determine
% the greatest common divisor of the polynomials \(a\) and \(b\), refer
% figure~\ref{euclidean}.
%
%
% \subsection[\texttt{\textbackslash polyfactorize}]
% {\normalfont\texttt{\textbackslash polyfactorize}\oarg{key=value list}\meta{polynomial}}
%
% The command prints a factorization of the polynomial as long as all except
% two roots are rational, see figure \ref{factorize}.
%
%
% \subsection{Low-level commands}
%
% To tell the whole truth, the commands above don't need the polynomials typed
% in verbatim. The internal representation of polynomials can be stored as
% replacement texts of control sequences and such control sequences can take
% the role of verbatim polynomials. This is also the case for \meta{\(a\)} and
% \meta{\(b\)} in table~\ref{low}, but each \meta{cs$_{\ldots}$} must be a
% control sequence, in which the result is saved.
%
% The command in table~\ref{low} can be used for low level calculations, and in
% particular to store polynomials for later use with the high-level commands.
% For example one could write the following.
% \begin{verbatim}
%    \polyadd\polya {(X^2+X+1)(X-1)-\frac\pi2}{0}% trick
%    \polymul\polyb {X-1}{1}             % another trick
%    Let's see how to divide \polyprint\polya{} by \polyprint\polyb.
%      \[\polylongdiv\polya\polyb\]\end{verbatim}
%
% \begin{table}
% \centering
% \begin{tabular}{r@{\enspace}ll}
% \meta{cs$_{a+b}$}&$\gets a+b$
%       & \cs{polyadd}\meta{cs$_{a+b}$}\meta{\(a\)}\meta{\(b\)}\\
%       &&\\
% \meta{cs$_{a-b}$}&$\gets a-b$
%       & \cs{polysub}\meta{cs$_{a-b}$}\meta{\(a\)}\meta{\(b\)}\\
%       &&\\
% \meta{cs$_{ab}$}&$\gets a\cdot b$
%       & \cs{polymul}\meta{cs$_{ab}$}\meta{\(a\)}\meta{\(b\)}\\
%       &&\\
% \meta{cs$_{a/b}$}&$\gets \lfloor a/b\rfloor$
%       & \cs{polydiv}\meta{cs$_{a/b}$}\meta{\(a\)}\meta{\(b\)}\\
% \cs{polyremainder}&$\gets a\bmod b$
%       &\\
%       &&\\
% \meta{cs$_{\gcd}$}&$\gets \gcd(a,b)$
%       & \cs{polygcd}\meta{cs$_{\gcd}$}\meta{\(a\)}\meta{\(b\)}\\
%       &&\\
% \multicolumn{2}{r}{print polynomial $a$}
%       & \cs{polyprint}\meta{\(a\)}\\
% \end{tabular}
% \caption{Low-level user commands}\label{low}
% \end{table}
%
%
% \section{Acknowledgments}
%
% I wish to thank
%   Ludger Humbert,
%   Karl Heinz Marbaise, and
%   Elke Niedermair
% for their tests and error reports.
%
%
% \StopEventually{^^A
% \begin{thebibliography}{1}
% \bibitem{TUGboat}
%   \textsc{Barbara Beeton} and \textsc{Donald Arseneau}.
%
%   \textit{Long division}.
%
%   In Jeremy Gibbons' \textit{Hey --- it works!},
%   TUGboat 18(2), June 1997, p.~75.
%
% \bibitem{calc}
%   \textsc{Kresten Krab Thorup}, \textsc{Frank Jensen}, and \textsc{Chris Rowley}.
%
%   \textit{The \texttt{calc} package, Infix notation arithmetic in \LaTeX}, 1998/07/07.
%
%   Available from \texttt{CTAN:} \texttt{macros/latex/required/tools}.
%
% \bibitem{keyval}
%   \textsc{David Carlisle}.
%
%   \textit{The \textsf{keyval} package}, 1999/03/16.
%
%   Available from \texttt{CTAN:} \texttt{macros/latex/required/graphics}.
%\end{thebibliography}}
%
%
% \CheckSum{4500}
%
%
% \section{Preliminaries}
%
% Let's start with identification.
%    \begin{macrocode}
%<*package>
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{polynom}[2017/07/17 0.19 (CH,HA)]
%    \end{macrocode}
% Now follow two frequently used definitions.
%
% \begin{macro}{\pld@AddTo}
% \begin{macro}{\pld@Extend}
% \meta{macro}\marg{contents}
% \begin{describe}
% adds \meta{contents} to the macro respectively does an |\expandafter| on the
% first token of \meta{contents} before doing so.
% \end{describe}
%    \begin{macrocode}
\def\pld@AddTo#1#2{\expandafter\def\expandafter#1\expandafter{#1#2}}
\def\pld@Extend#1#2{%
    \expandafter\pld@AddTo\expandafter#1\expandafter{#2}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@ExpandTwo}
% expands the respectively first tokens of |#2| and |#3| and puts all as
% argument after |#1|. Note that |#2| and |#3| need not to be single tokens.
%    \begin{macrocode}
\def\pld@ExpandTwo#1#2#3{%
    \expandafter\def\expandafter\pld@temp\expandafter{#2}%
    \pld@Extend\pld@temp{#3}%
    \expandafter#1\pld@temp}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@if}
% is used as a temporary and local switch.
%    \begin{macrocode}
\def\pld@true{\let\pld@if\iftrue}
\def\pld@false{\let\pld@if\iffalse}
\pld@false
%    \end{macrocode}
% \end{macro}
%
%
% \section{The user interface}
%
% \begin{macro}{\polyset}
% This command just `inserts' the family name |pld| and requires the
% \packagename{keyval} package.
%    \begin{macrocode}
\RequirePackage{keyval}[1997/11/10]
\def\polyset{\setkeys{pld}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@IfVar}
% The variables are stored in a comma separated list. Here we look after
% |#1| being an element and execute the second argument \meta{then} or the
% third argument \meta{else}.
%    \begin{macrocode}
\def\pld@IfVar#1{%
    \def\pld@temp##1,#1,##2##3\relax{%
        \ifx\@empty##3\@empty \expandafter\@secondoftwo
                        \else \expandafter\@firstoftwo \fi}%
    \expandafter\pld@temp\pld@variables,#1,\@empty\relax}
%    \end{macrocode}
% \end{macro}
% The key iterates down the tokens and expands the list, making a new
% key for every variable.
%    \begin{macrocode}
\define@key{pld}{vars}{%
  \let\pld@variables\@empty
  \@tfor\pld@temp:=#1\do{%
    \pld@Extend\pld@variables{\expandafter,\pld@temp}%
    \edef\pld@temp{%
      \noexpand\define@key{pld}{\pld@temp}%
      {\noexpand\pld@GetValue{\pld@temp}{####1}}%
    }%
    \pld@temp
  }%
}
%    \end{macrocode}
% \begin{macro}{\pld@GetValue}
% Helper macro to retrieve the value of a variable.
%    \begin{macrocode}
\def\pld@GetValue#1#2{%
  \pld@GetPoly{\pld@polya}{}{#2}%
  \ifx\pld@polya\@empty\def\pld@polya{\pld@R 01}\fi
  \expandafter\let\csname pld@value@#1\endcsname\pld@polya
}
%    \end{macrocode}
%    \begin{macrocode}
\polyset{vars=Xx}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@iftopresult}
% determines the printing style for long divisions. The key checks for the
% macro definition |\pld@style|\meta{name}, \ldots
%    \begin{macrocode}
\define@key{pld}{style}
    {\@ifundefined{pld@style#1}%
         {\PackageError{polynom}{Unknown style `#1'}%
          {Arguments can be `A' or `B' or `C' or `D'.}}%
         {\let\pld@style=#1%
          \@nameuse{pld@style#1}}}
%    \end{macrocode}
% which are defined here.
%    \begin{macrocode}
\def\pld@styleA{\let\pld@iftopresult\iftrue}
\def\pld@styleB{\let\pld@iftopresult\iffalse}
\let\pld@styleC\pld@styleB
\let\pld@styleD\pld@styleB
%    \end{macrocode}
%    \begin{macrocode}
\polyset{style=A}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@leftdelim}
% \begin{macro}{\pld@rightdelim}
% \begin{macro}{\pld@leftxdelim}
% \begin{macro}{\pld@rightxdelim}
% We make left and right delimiters definable.
%    \begin{macrocode}
\define@key{pld}{delims}
    {\@ifnextchar[\pld@delims
                  {\pld@delims[{}{}]}#1{}{}}
\def\pld@delims[#1#2]#3#4{%
    \def\pld@leftxdelim{#1}\def\pld@rightxdelim{#2}%
    \def\pld@leftdelim{#3}\def\pld@rightdelim{#4}}
\polyset{delims=[{\left.}{\right.}]{\left(}{\right)}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@div}
% Moreover one can customize the division sign for the C style.
%    \begin{macrocode}
\define@key{pld}{div}{\def\pld@div{#1}}
\polyset{div=\div}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@stage}
% \begin{macro}{\pld@currstage}
% Ensure a positive value.
%    \begin{macrocode}
\define@key{pld}{stage}{%
    \@tempcnta#1\relax \ifnum\@tempcnta<\@ne \@tempcnta\@ne \fi
    \edef\pld@stage{\the\@tempcnta}}
%    \end{macrocode}
%    \begin{macrocode}
\newcount\pld@currstage
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% The following definitions all have the same scheme: get the polynomial(s),
% do the operation, and assign or print the result. And they all use macros
% which are defined in later sections.
%
% \begin{macro}{\polymul}
% Just the things stated.
%    \begin{macrocode}
\newcommand*\polymul[1]{%
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\pld@MultiplyPoly#1\pld@polya\pld@polyb
                 \ignorespaces}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polydiv}
% Ditto.
%    \begin{macrocode}
\newcommand*\polydiv[1]{%
    \begingroup
    \let\pld@stage\maxdimen
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\pld@DividePoly\pld@polya\pld@polyb
                 \let#1\pld@quotient
                 \let\polyremainder\pld@remainder
                 \pld@RestoreAftergroup#1\polyremainder\relax
    \endgroup\ignorespaces}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polylongdiv}
% Ditto. We use an if to use the fix by Hendrik Vogt only inside this macro.
%    \begin{macrocode}
\newif\ifpld@InsidePolylongdiv
\newcommand*\polylongdiv[1][]{%
    \begingroup\pld@InsidePolylongdivtrue
    \let\pld@stage\maxdimen \polyset{#1}%
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\pld@LongDividePoly\pld@polya\pld@polyb
                 \pld@PrintLongDiv
    \endgroup \ignorespaces}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polylonggcd}
% Ditto.
%    \begin{macrocode}
\newcommand*\polylonggcd[1][]{%
    \begingroup
    \let\pld@stage\maxdimen \polyset{#1}%
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\pld@LongEuclideanPoly\pld@polya\pld@polyb
                 \pld@PrintLongEuclidean
    \endgroup \ignorespaces}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polygcd}
% Ditto.
%    \begin{macrocode}
\newcommand*\polygcd[1]{%
    \begingroup
    \let\pld@stage\maxdimen
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\pld@LongEuclideanPoly\pld@polya\pld@polyb
                 \global\let\@gtempa\pld@vb
    \endgroup \let#1\@gtempa \ignorespaces}}
%    \end{macrocode}
% A bug report by Elke Niedermair ^^A {e.a.n@gmx.de}{2002/10/29}{undefined control sequence \pld@stage}
% led to the initialization of |\pld@stage| -- and the surrounding
%|\begingroup| and |\endgroup|.
% \end{macro}
%
% \begin{macro}{\polyfactorize}
% Ditto.
%    \begin{macrocode}
\newcommand*\polyfactorize{%
    \pld@GetPoly\pld@current
                {\pld@Factorize\pld@current \ensuremath{\pld@allines}}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polyprint}
% Ditto.
%    \begin{macrocode}
\newcommand*\polyprint{%
    \pld@GetPoly{\pld@polya}%
                {\ensuremath{\pld@PrintPoly\pld@polya}}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polyadd}
% Get the polynomials, add them via appending the representation, and
% normalize the result via simplification.
%    \begin{macrocode}
\newcommand*\polyadd[1]{%
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\let#1\pld@polya \pld@ExtendPoly#1\pld@polyb
                 \pld@Simplify#1%
                 \ignorespaces}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\polysub}
% Ditto.
%    \begin{macrocode}
\newcommand*\polysub[1]{%
    \pld@GetPoly{\pld@polya\pld@polyb}%
                {\def\pld@tempoly{\pld@R{-1}1}%
                 \pld@MultiplyPoly\pld@polyb\pld@polyb\pld@tempoly
                 \let#1\pld@polya \pld@ExtendPoly#1\pld@polyb
                 \pld@Simplify#1%
                 \ignorespaces}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@RestoreAftergroup}
% We just iterate down the control sequences, add |\def#1{|\meta{contents of
% \texttt{\#1}}|}| to |\@gtempa|, and execute it \cs{aftergroup}.
%    \begin{macrocode}
\def\pld@RestoreAftergroup{%
    \global\let\@gtempa\@empty
    \pld@RestoreAfter@}
\def\pld@RestoreAfter@#1{%
    \ifx\relax#1%
        \aftergroup\@gtempa
    \else
        \global\pld@Extend\@gtempa{\expandafter\def\expandafter#1%
                                   \expandafter{#1}}%
        \expandafter\pld@RestoreAfter@
    \fi}
%    \end{macrocode}
% \end{macro}
%
%
% \section{Internal data format}\label{sInternalDataFormat}
%
% A polynomial is a finite sum $\meta{$\mathrm{monomial}_1$}+\ldots+
% \meta{$\mathrm{monomial}_n$}$ of monomials. In the internal data format, the
% monomials will be sorted by degree---in the multivariate case not by the
% total degree but by the degree of the first variable, then by the degree of
% the second variable, and so on.
%
% Each monomial is a product of \emph{rationals}, general \emph{fractions},
% \emph{symbolic factors}, and \emph{variables}. The factors are represented
% in the following format.
% \begin{enumerate}
% \item |\pld@R|\marg{integer nominator}\marg{integer denominator} for rationals,
% \item |\pld@F|\marg{nominator}\marg{denominator} for general fractions,
% \item |\pld@S|\marg{symbol}\marg{exponent} for symbolic factors, and
% \item |\pld@V|\marg{symbol}\marg{exponent} for variables.
% \end{enumerate}
% As a special case \meta{nominator} and/or \meta{denominator} may be empty,
% which stands for a factor $1={}$|\pld@R{1}{1}|.
%
% \begin{table}[tp]
% \begin{tabular}{cl}
% \emph{mathematians write}&\multicolumn{1}{c}{\emph{internal representation}}\\[1ex]
% $X^2-1$      & |\pld@V{X}{2}+\pld@R{-1}{1}|\\[1ex]
% $\frac1e X$  & |\pld@F{\pld@R{1}{1}}{\pld@S{e}{1}}\pld@V{X}{1}|\\
%              & |\pld@S{e}{-1}\pld@V{X}{1}|\\[1ex]
% $\frac1{10}$ & |\pld@F{\pld@R{1}{1}}{\pld@R{10}{1}}|\\
%              & |\pld@R{1}{10}|
% \end{tabular}
% \caption{Mathematical notation versus internal representation}\label{mvi}
% \end{table}
% Table \ref{mvi} shows examples of the internal data format. As you can see,
% sometimes there are various ways to represent the same polynomial. The
% exact internal data depends on how you enter the factors and which state has
% been reached in the division algorithm, for example.
%
% And now some definitions which work on representations of polynomials, first
% macros to `look at' polynomials and then macros to build them.
%
% \begin{macro}{\pld@SplitMonom}
% \meta{|\#1\#2| submacro}\marg{monomial representation}
% \begin{describe}
% calls the given macro with the `nonvariable' part as first and the `variable'
% part as second argument. Each of them can be empty. Note that this definition
% makes an assumption on the order of the factors in the representation, namely
% that the variable part comes at the end.
% \end{describe}
%    \begin{macrocode}
\def\pld@SplitMonom#1#2{%
    \pld@SplitMonom@#2\pld@V\relax {\pld@SplitMonom@V#1#2\relax}%
                                   {#1{#2}{}}}
\def\pld@SplitMonom@#1\pld@V#2\relax{%
    \ifx\@empty#2\@empty \expandafter\@secondoftwo
                   \else \expandafter\@firstoftwo \fi}
\def\pld@SplitMonom@V#1#2\pld@V#3\relax{#1{#2}{\pld@V#3}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@SplitMonomS}
% \meta{|\#1\#2| submacro}\marg{`monomial representation'}
% \begin{describe}
% does the same but splits at |\pld@S| to separate numerals and symbols.
% \end{describe}
%    \begin{macrocode}
\def\pld@SplitMonomS#1#2{%
    \pld@SplitMonomS@#2\pld@S\relax {\pld@SplitMonomS@S#1#2\relax}%
                                   {#1{#2}{}}}
\def\pld@SplitMonomS@#1\pld@S#2\relax{%
    \ifx\@empty#2\@empty \expandafter\@secondoftwo
                   \else \expandafter\@firstoftwo \fi}
\def\pld@SplitMonomS@S#1#2\pld@S#3\relax{#1{#2}{\pld@S#3}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@IfSum}
% \marg{polynomial representation}\marg{then}\marg{else}
% \begin{describe}
% executes \meta{then} if and only if the polynomial is a sum (of more than
% one monomial).
% \end{describe}
%    \begin{macrocode}
\def\pld@IfSum#1{\pld@IfSum@#1+\@empty+\relax+}
\def\pld@IfSum@#1+#2+\relax+{%
    \ifx\@empty#2\@empty \expandafter\@secondoftwo
                   \else \expandafter\@firstoftwo \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@DefNegative}
% \meta{macro}\marg{monomial representation}
% \begin{describe}
% negates the monomial and puts it into the macro.
% \end{describe}
%    \begin{macrocode}
\def\pld@DefNegative#1#2{\pld@DefNegative@#1#2\@empty}
\def\pld@DefNegative@#1#2#3#4#5\@empty{%
    \ifx #2\pld@R\def#1{\pld@R{-#3}{#4}#5}%
           \else \def#1{\pld@R{-1}1#2{#3}{#4}#5}\fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@DefInverse}
% \meta{inverse macro}\marg{monomial representation}
% \begin{describe}
% puts a representation of the monomials' reciprocal into the macro.
% \end{describe}
%    \begin{macrocode}
\def\pld@DefInverse#1#2{%
    \let#1\@empty
    \pld@DefInverse@#1#2\relax\@empty\@empty}
%    \end{macrocode}
% Here we just interchange nominator and denominator or negate the exponent.
%    \begin{macrocode}
\def\pld@DefInverse@#1#2#3#4{%
    \ifx\relax#2\relax \expandafter\@gobbletwo \else
        \ifx #2\pld@R \pld@AddTo#1{\pld@R{#4}{#3}}\else
        \ifx #2\pld@F \pld@AddTo#1{\pld@F{#4}{#3}}\else
        \ifx #2\pld@S \pld@AddTo#1{\pld@S{#3}{-#4}}\else
        \ifx #2\pld@V \pld@AddTo#1{\pld@V{#3}{-#4}}\else
            \pld@DIError
        \fi \fi \fi \fi
    \fi
    \pld@DefInverse@#1}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AddToPoly}
% \begin{macro}{\pld@ExtendPoly}
% \meta{polynomial}\marg{monomial}
% \begin{describe}
% adds \meta{monomial} as a new summand to \meta{polynomial} or does an
% |\expandafter| on the first token before doing so.
% \end{describe}
%    \begin{macrocode}
\def\pld@AddToPoly#1#2{%
    \ifx #1\@empty \def#1{#2}\else
                   \pld@AddTo#1{+#2}\fi}
\def\pld@ExtendPoly#1#2{%
    \ifx #1\@empty \pld@Extend#1{#2}\else
    \ifx #2\@empty
             \else \pld@Extend#1{\expandafter+#2}\fi \fi}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@R}
% \begin{macro}{\pld@F}
% \begin{macro}{\pld@S}
% \begin{macro}{\pld@V}
% These macros just contain distinct single characters. We will change the
% definitions locally when we output a polynomial, for example.
%    \begin{macrocode}
\def\pld@R{r}
\def\pld@F{f}
\def\pld@S{s}
\def\pld@V{v}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
%
% \section{Scanning input}
%
% \begin{macro}{\pld@GetPoly}
% |{|\meta{macro$_1$}\ldots\meta{macro$_k$}|}| \marg{do after} \marg{polynomial$_1$}\ldots\marg{polynomial$_k$}
% \begin{describe}
% This definition parses all user supplied polynomials. For $i=1,\ldots,k$,
% it assigns the internal representation of \meta{polynomial$_i$} to
% \meta{macro$_i$} and executes \meta{do after}wards. `\marg{polynomial$_i$}'
% may be a stored polynomial---that means a control sequence---in which case
% the argument braces \emph{must} be omitted.
% \end{describe}
% First we initialize data and check whether the user provides an explicit
% polynomial.
%    \begin{macrocode}
\def\pld@GetPoly#1#2{%
    \def\pld@pool{#1}\def\pld@aftermacro{#2}%
    \pld@GetPoly@}
\def\pld@GetPoly@{%
    \@ifnextchar\bgroup \pld@GetPolyArg\pld@GetPolyLet}
%    \end{macrocode}
% Such a polynomial is scanned and then assigned to a macro from the pool.
%    \begin{macrocode}
\def\pld@GetPolyArg#1{%
    \pld@Scan{#1}%
    \pld@GetPolyLet\pld@tempoly}
%    \end{macrocode}
% Here we get one macro from the pool, assign the polynomial, and continue if
% the pool isn't empty.
%    \begin{macrocode}
\def\pld@GetPolyLet{\expandafter\pld@GetPolyLet@\pld@pool\relax}
\def\pld@GetPolyLet@#1#2\relax#3{%
    \let#1#3\def\pld@pool{#2}%
    \pld@Simplify#1%
    \ifx\pld@pool\@empty \expandafter\pld@aftermacro
                   \else \expandafter\pld@GetPoly@ \fi}
%    \end{macrocode}
% \end{macro}
%
% Now we actually scan the input. Section \emph{4 The evaluation scheme} of
% the \texttt{calc} package \cite{calc} explains how this is done in general.
% Together with the implementation part of the that package, it's an excellent
% introduction---if you are familiar with \TeX. However, some things are
% different:
% \begin{itemize}
% \item We additionally detect |\frac|, |\cdot|, symbols, and variables.
% \item To write $XY$ instead of $X\cdot Y$ or $X*Y$, we provide an implicit
%       multiplication.
% \item |\pld@ScanIt| does what |\calc@pre@scan| and |\calc@post@scan| do.
% \item In terms of the \texttt{calc} package, we clear the local `register B'
%       |\pld@tempoly| after each |\begingroup| (for providing a faster
%       multiplication, see below) and we assign the value of that register
%       to the global `register A' |\@gtempa| before each |\endgroup|.
% \item Multiplication with a single factor (symbol, variable, fraction,
%       number) makes group level changes only if the current term is a sum.
%       Otherwise it just adds the factor to |\pld@tempoly|. This saves many
%       multiplications of polynomials.
% \end{itemize}
% We begin with basic definitions.
%
% \begin{macro}{\pld@ScanBegingroup}
% \begin{macro}{\pld@ScanEndgruop}
% Just what was stated above.
%    \begin{macrocode}
\def\pld@ScanBegingroup{\begingroup \let\pld@tempoly\@empty}
\def\pld@ScanEndgroup{\pld@ScanSetA \endgroup}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@ScanSetA}
% \begin{macro}{\pld@ScanSetB}
% In the \texttt{calc} package the second routine is called |\scan@initB| (but
% with registers instead of macros, of course)
%    \begin{macrocode}
\def\pld@ScanSetA{\global\let\@gtempa\pld@tempoly}
\def\pld@ScanSetB{\let\pld@tempoly\@gtempa}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@Scan}
% corresponds to |\calc@assign@generic|.
%    \begin{macrocode}
\def\pld@Scan#1{%
    \let\pld@tempoly\@empty
      \pld@ScanOpen(#1\relax
    \pld@ScanEndgroup \pld@ScanEndgroup}
%    \end{macrocode}
% The brake |\relax| terminates the main scanning loop.
%    \begin{macrocode}
\def\pld@ScanIt#1{%
    \ifx \relax#1\let\pld@next\@gobble \else
%    \end{macrocode}
% The tokens |+|, |-|, |*|, |\cdot|, |/|, and |)| let us make the according
% translations.
%    \begin{macrocode}
    \ifx +#1\let\pld@next\pld@ScanAdd\else
    \ifx -#1\let\pld@next\pld@ScanSubtract\else
    \ifx *#1\let\pld@next\pld@ScanMultiply\else
    \ifx \cdot#1\let\pld@next\pld@ScanMultiply\else
    \ifx /#1\let\pld@next\pld@ScanDivide\else
    \ifx )#1\let\pld@next\pld@ScanClose\else
    \ifx ^#1\let\pld@next\pld@ScanPower\else
%    \end{macrocode}
% Other tokens are `preceeded' by an implicit multiplication, as you will see
% below.
%    \begin{macrocode}
        \ifx \frac#1\let\pld@next\pld@ScanFrac\else
        \ifx (#1\let\pld@next\pld@ScanOpen\else
%    \end{macrocode}
% Now we check for a digit and a variable, respectively. If none of them is
% given, we treat the argument as a symbol.
%    \begin{macrocode}
            \pld@IfNumber{#1}%
                {\let\pld@next\pld@ScanNumeric}%
                {\pld@IfVar{#1}{\let\pld@next\pld@ScanVar}%
                               {\let\pld@next\pld@ScanSymbol}}%
        \fi \fi
    \fi \fi \fi \fi \fi \fi \fi \fi
    \pld@next #1}
%    \end{macrocode}
% For speedy scanning we could alternatively define
% \begin{verbatim}
%    \def\pld@ScanIt#1{%
%        \expandafter\let\expandafter\pld@next\csname
%                                 pld@Scan@\string#1\endcsname
%        \ifx\relax\pld@next
%            \pld@IfVar{#1}{\let\pld@next\pld@ScanVar}%
%                          {\let\pld@next\pld@ScanSymbol}%
%        \fi
%        \pld@next #1}\end{verbatim}
% and make appropriate definitions |\pld@Scan@\relax| (one single control
% sequence), |\pld@Scan@|$\langle$\texttt{+$\vert$-$\vert$*$\vert$\bslash
% cdot$\vert$/$\vert$)$\vert$\textasciicircum$\vert$\bslash frac$\vert$($\vert
% $0$\vert\ldots\vert$9}$\rangle$ instead of |\pld@ScanAdd| and all the other
% definitions below.
% \end{macro}
%
% \begin{macro}{\pld@IfNumber}
% execute the first or second argument depending on whether |#1| is found in
% |0123456789|.
%    \begin{macrocode}
\def\pld@IfNumber#1{%
    \def\pld@temp##1#1##2##3\relax{%
        \ifx\@empty##2\@empty \expandafter\@secondoftwo
                        \else \expandafter\@firstoftwo \fi}%
    \pld@temp 0123456789#1\@empty\relax}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@ScanOpen}
% \begin{macro}{\pld@ScanClose}
% correspond to |\calc@open| and |\calc@close|. A left parentheses implicitly
% inserts a multiplication in many (or even most) cases since the parenthesized
% expression must be viewed as a potential sum.
% The simple |\pld@ScanImplicitMultiply| in version 0.11 didn't insert a
% multiplication when scanning |2(X+1)|, for example.
%    \begin{macrocode}
\def\pld@ScanOpen({%
    \ifx\@empty\pld@tempoly\else
        \pld@ScanMultiplyBase\pld@ScanBbyA
    \fi
    \pld@ScanBegingroup \aftergroup\pld@ScanSetB
    \pld@ScanBegingroup \aftergroup\pld@ScanSetB
    \pld@ScanIt}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@ScanClose){%
    \pld@ScanEndgroup \pld@ScanEndgroup \pld@ScanSetA
    \pld@ScanIt}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@ScanAdd}
% \begin{macro}{\pld@ScanSubtract}
% correspond to |\calc@add| and |\calc@subtract|.
%    \begin{macrocode}
\def\pld@ScanAdd+{\pld@ScanAddBase\pld@ScanAtoB}
\def\pld@ScanSubtract-{\pld@ScanAddBase\pld@ScanAfromB}
\def\pld@ScanAddBase#1{%
    \pld@ScanEndgroup \pld@ScanEndgroup
    \pld@ScanBegingroup \aftergroup#1%
    \pld@ScanBegingroup \aftergroup\pld@ScanSetB
    \pld@ScanIt}
%    \end{macrocode}
% For a subtraction we just add a factor |\pld@R{-1}{1}|.
%    \begin{macrocode}
\def\pld@ScanAtoB{\pld@ExtendPoly\pld@tempoly\@gtempa}
\def\pld@ScanAfromB{\pld@ExtendPoly\pld@tempoly{\@gtempa\pld@R{-1}1}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@ScanMultiply}
% \begin{macro}{\pld@ScanDivide}
% instead of |\calc@multiply| and |\calc@divide|.
%    \begin{macrocode}
\def\pld@ScanMultiply#1{\pld@ScanMultiplyBase\pld@ScanBbyA \pld@ScanIt}
\def\pld@ScanDivide/{\pld@ScanMultiplyBase\pld@ScanDivBbyA \pld@ScanIt}
\def\pld@ScanMultiplyBase{%
    \pld@ScanEndgroup \pld@ScanBegingroup \aftergroup}
%    \end{macrocode}
% Division is here adding a fraction, thus this is limited to numbers and
% symbols. No variables should appear in the expression after |/|.
%    \begin{macrocode}
\def\pld@ScanBbyA{\pld@MultiplyPoly\pld@tempoly\pld@tempoly\@gtempa}
\def\pld@ScanDivBbyA{%
    \def\pld@temp{\pld@F{}}%
    \pld@Extend\pld@temp{\expandafter{\@gtempa}}%
    \pld@Extend\pld@tempoly\pld@temp}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@ScanPower}
% We calculate the |#1|-th power of |\pld@tempoly| by successive
% multiplication. Note that this code---as most code of this package---is not
% optimized for speed.
%    \begin{macrocode}
\def\pld@ScanPower^#1{%
    \let\pld@polya\pld@tempoly
    \@multicnt#1\relax
    \loop \ifnum\@multicnt>\@ne
        \advance\@multicnt\m@ne
        \pld@MultiplyPoly\pld@tempoly\pld@tempoly\pld@polya
    \repeat
    \pld@ScanIt}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@ScanImplicitMultiply}
% inserts a multiplication if and only if the local register B is a sum.
%    \begin{macrocode}
\def\pld@ScanImplicitMultiply{%
    \expandafter\pld@IfSum\expandafter{\pld@tempoly}%
        {\pld@ScanMultiplyBase\pld@ScanBbyA}%
        {}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@ScanNumeric}
% We assign the integer to a count register and issue an error if it's
% fractional.
%    \begin{macrocode}
\def\pld@ScanNumeric{%
    \pld@ScanImplicitMultiply
    \let\pld@temp\frac \let\frac\relax
    \afterassignment\pld@ScanNumeric@ \@tempcnta}
\def\pld@ScanNumeric@#1{%
    \let\frac\pld@temp
    \ifx #1.%
        \PackageError{polynom}{noninteger constants not supported}%
        {Constants must be integers in TeX's word range.\MessageBreak
         The fractional part will be lost.}%
        \def\pld@next##1{\afterassignment\pld@ScanIt\@tempcnta}%
    \else
        \let\pld@next\pld@ScanIt
    \fi
%    \end{macrocode}
% We add the integer to the polynomial and continue the scan.
%    \begin{macrocode}
    \pld@Extend\pld@tempoly
        {\expandafter\pld@R\expandafter{\the\@tempcnta}1}%
    \pld@next #1}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@ScanVar}
% \begin{macro}{\pld@ScanSymbol}
% are defined in terms of a submacro.
%    \begin{macrocode}
\def\pld@ScanVar{\pld@ScanImplicitMultiply \pld@ScanVS\pld@V}
\def\pld@ScanSymbol{\pld@ScanImplicitMultiply \pld@ScanVS\pld@S}
%    \end{macrocode}
% The submacro checks all super- and subscript variations and adds
% the data to the polynomial. The suffixes |u| and |l| stand for
% upper and lower.
%    \begin{macrocode}
\def\pld@ScanVS#1#2{%
    \@ifnextchar^{\pld@ScanVS@u#1{#2}}%
                 {\@ifnextchar_{\pld@ScanVar@l#1{#2}}%
                               {\pld@AddTo\pld@tempoly{#1{#2}1}%
                                \pld@ScanIt}}}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@ScanVS@u#1#2^#3{%
    \@ifnextchar_{\pld@ScanVS@ul#1{#2}{#3}}%
                 {\pld@AddTo\pld@tempoly{#1{#2}{#3}}\pld@ScanIt}}
\def\pld@ScanVS@l#1#2_#3{%
    \@ifnextchar^{\pld@ScanVS@lu#1{#2}{#3}}%
                 {\pld@AddTo\pld@tempoly{#1{#2_{#3}}1}\pld@ScanIt}}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@ScanVS@ul#1#2#3_#4{%
    \pld@AddTo\pld@tempoly{#1{#2_{#4}}{#3}}\pld@ScanIt}
\def\pld@ScanVS@lu#1#2#3^#4{%
    \pld@AddTo\pld@tempoly{#1{#2_{#3}}{#4}}\pld@ScanIt}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@ScanFrac}
% A fraction scans nominator and denominator seperately and add them to the
% current |\pld@tempoly|.
%    \begin{macrocode}
\def\pld@ScanFrac#1#2#3{%
    \pld@ScanImplicitMultiply
    \begingroup
      \pld@Scan{#2}\pld@AddTo\pld@tempoly\relax
      \global\let\@gtempa\pld@tempoly
    \endgroup
    \pld@Extend\pld@tempoly{\expandafter\pld@F\expandafter{\@gtempa}}%
%    \end{macrocode}
%    \begin{macrocode}
    \begingroup
      \pld@Scan{#3}\pld@AddTo\pld@tempoly\relax
      \global\let\@gtempa\pld@tempoly
    \endgroup
    \pld@Extend\pld@tempoly{\expandafter{\@gtempa}}%
%    \end{macrocode}
%    \begin{macrocode}
    \pld@ScanIt}
%    \end{macrocode}
% \end{macro}
%
%
% \section{Basic printing}
%
% \begin{macro}{\pld@PrintPoly}
% \meta{polynomial macro}
% \begin{describe}
% prints the polynomial represented by the macro contents. An empty macro
% stands for `0'.
% \end{describe}
% The implementation follows that description and uses |\pld@PrintMonoms|.
%    \begin{macrocode}
\def\pld@PrintPolyArg#1{%
    \def\pld@temp{#1}\pld@PrintPoly\pld@temp}
\def\pld@PrintPoly#1{%
    \ifx\@empty#1\@empty 0\else
        \pld@firsttrue \expandafter\pld@PrintMonoms#1+\relax+%
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintPolyWithDelims}
% \meta{polynomial macro}
% \begin{describe}
% prints the polynomial represented by the macro contents. The polynomial is
% enclosed in the user defined delimiters except when the polynomial is a
% single summand. Then just that summand is printed.
% \end{describe}
% According to the result of |\pld@IfSum| we insert the delimiters.
%    \begin{macrocode}
\def\pld@PrintPolyWithDelimsArg#1{%
    \def\pld@temp{#1}\pld@PrintPolyWithDelims\pld@temp}
\def\pld@PrintPolyWithDelims#1{%
    \ifx\@empty#1\@empty 0\else
        \pld@firsttrue
        \expandafter\pld@IfSum\expandafter{#1}\pld@true\pld@false
        \pld@if \pld@leftdelim
                \expandafter\pld@PrintMonoms#1+\relax+%
                \pld@rightdelim
          \else \expandafter\pld@PrintMonoms#1+\relax+\fi
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintInit}
% is called before we print a factor of a monomial. First we have to `reset'
% |\pld@R| to its primary definition. Its special definition below suppresses
% the output of a factor `1' if this factor is not required.
%    \begin{macrocode}
\def\pld@PrintInit{%
    \let\pld@R\pld@PrintRational \strut
%    \end{macrocode}
% The switch |\pld@iffirst| indicates whether we are working on the first
% summand of a polynomial, that means whether or not we can omit a plus.
% According to the value of the accumulator, we print its sign and/or value.
%    \begin{macrocode}
    \pld@AccuIfNegative
       {\pld@AccuNegate \pld@iffirst\pld@minustrue\else{}\fi -}%
                       {\pld@iffirst\pld@minusfalse\else{}+\fi}%
    \pld@firstfalse
    \pld@AccuIfAbsOne{}{\pld@AccuPrint \pld@true}%
%    \end{macrocode}
% The switch |\pld@if| is set true if and only if we have printed something
% for the monomial (except the sign). So we know when to insert the omitted
% factor `1'.
%
% At the end of |\pld@PrintInit|, the macro throws away itself since all the
% `initialisation' done here is necessary only once for a summand. Note that
% this is done inside a group below, thus the meaning is not lost.
%    \begin{macrocode}
    \let\pld@PrintInit\@empty}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@minustrue{\global\let\pld@ifminus\iftrue}
\def\pld@minusfalse{\global\let\pld@ifminus\iffalse}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@iffirst}
% from above.
%    \begin{macrocode}
\def\pld@firsttrue{\global\let\pld@iffirst\iftrue}
\def\pld@firstfalse{\global\let\pld@iffirst\iffalse}
\pld@firstfalse
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintMonoms}
% While not reaching the end of the polynomial, the accumulator gets `1' and
% we redefine |\pld@R|,\ldots,|\pld@V|. For example, a rational is just saved
% by multiplying it with the current accumulator.
%    \begin{macrocode}
\def\pld@PrintMonoms#1+{%
    \ifx\relax#1\else
        \begingroup
          \pld@AccuSetX11%
          \let\pld@R\pld@AccuMul
          \let\pld@F\pld@PrintFrac
          \let\pld@S\pld@PrintSymbol
          \let\pld@V\pld@PrintSymbol
%    \end{macrocode}
% Then we indicate that nothing has been printed so far, print the factors (if
% any) by executing the code, and finally print the accumulator if necessary.
%    \begin{macrocode}
          \pld@false
          #1%
          \pld@PrintInit
          \pld@if\else \pld@AccuPrint \fi
        \endgroup
        \expandafter\pld@PrintMonoms
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintRational}
% is |\pld@R|: indicate that we print something, load the accumulator, and
% print it. Note that we don't need to call |\pld@PrintInit| since it has
% already been done!
%    \begin{macrocode}
\def\pld@PrintRational#1#2{%
    \pld@true \pld@AccuSetX{#1}{#2}\pld@AccuPrint}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintSymbol}
% is |\pld@S| or |\pld@V|: init, indicate, and print the symbol with exponent.
%    \begin{macrocode}
\def\pld@PrintSymbol#1#2{%
    \pld@PrintInit \pld@true #1\ifnum#2=\@ne\else^{#2}\fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintFrac}
% is |\pld@F|: init, indicate, and print the fraction \ldots\space no, wait!
% First we check whether the denominator is `1'. In that case we don't use a
% fraction; we enclose the nominator in delimiters if necessary.
%    \begin{macrocode}
\def\pld@PrintFrac#1#2{%
    \pld@PrintInit \pld@true
    \ifx\@empty#2\@empty
        \begingroup
          \pld@IfSum{#1}\pld@true\pld@false
          \pld@if\pld@leftdelim #1\pld@rightdelim\else #1\fi
        \endgroup
        \expandafter\@gobbletwo
    \else
        \expandafter\pld@PrintFrac@
    \fi
    {#1}{#2}}
%    \end{macrocode}
% Otherwise we check the nominator.
%    \begin{macrocode}
\def\pld@PrintFrac@#1#2{%
    \ifx\@empty#1\@empty \frac1{#2}\else \frac{#1}{#2}\fi}
%    \end{macrocode}
% \end{macro}
%
% These printing routines do \emph{not} handle all representations which are
% legal in the sense of section \ref{sInternalDataFormat}. For example,
% |\pld@R{1}{1}\pld@V{X}{1}\pld@R{-1}{1}| would be printed as $X-1$.
% A correct output is guaranteed if there is at most one rational at the
% very beginning of the representation. Thus we normalize the internal data:
% condense, sort and simplify factors and summands. This will keep us busy
% for the next (p)ages.
%
%
% \section{Simplifying}
%
%
% \subsection{Phase I: Condense factors of summands}
%
% \begin{macro}{\pld@CondenseFactors}
% \meta{polynomial macro}
% \begin{describe}
% Here we condense rationals, fractions, and the exponents of symbolic factors
% and variables of each monomial in \meta{polynomial macro}. Afterwards the
% factors appear exactly in this order: if present a rational number comes
% first, then fractions if any, then symbols, and finally the variables.
% For example, the representation of
% {\makeatletter\pld@Scan{\frac23\frac1e5X^3\frac32X}
%         $\pld@PrintPoly\pld@tempoly$
% becomes  \pld@CondenseFactors\pld@tempoly
%         $\pld@PrintPoly\pld@tempoly$}.
% \end{describe}
% As for printing, we redefine |\pld@R|,\ldots,|\pld@V| and iterate through
% the monomials.
%    \begin{macrocode}
\def\pld@CondenseFactors#1{%
    \ifx\@empty#1\else
        \begingroup
          \let\pld@R\pld@AccuMul
          \let\pld@F\pld@CondenseFrac
          \let\pld@S\pld@CFSymbol
          \let\pld@V\pld@CFVar
%    \end{macrocode}
% The following two assignments allow |#1| to be |\pld@temp| or |\pld@tempoly|.
%    \begin{macrocode}
          \let\pld@temp#1\let\pld@tempoly\@empty
          \expandafter\pld@CF@loop\pld@temp+\relax+%
          \global\let\@gtempa\pld@tempoly
        \endgroup
        \let#1\@gtempa
    \fi}
%    \end{macrocode}
% For each monomial the factors are kept in separate `registers': rationals in
% the accumulator, fractions in |\pld@frac|, symbols in |\pld@symbols|, and
% variables in |\pld@vars|. |\pld@if| is set true if and only if there is a
% `general' fraction.
%    \begin{macrocode}
\def\pld@CF@loop#1+{%
    \ifx\relax#1\else
        \begingroup
          \pld@AccuSetX11%
          \def\pld@frac{{}{}}\let\pld@symbols\@empty\let\pld@vars\@empty
          \pld@false
          #1%
          \let\pld@temp\@empty
%    \end{macrocode}
% Now we put together the collected data (if the rational constant is not
% zero): If the rational constant does not equal one, we place it in front of
% all other factors.
%    \begin{macrocode}
          \pld@AccuIfZero{}%
          {\pld@AccuIfOne{}{\pld@AccuGet\pld@temp
                            \edef\pld@temp{\noexpand\pld@R\pld@temp}}%
%    \end{macrocode}
% Then follow fractions, symbols and variables.^^A
% \footnote{This is the right place to \emph{simplify} general fractions and
%       symbols. Here we are in the special case that they don't contain
%       any rationals as `over all' factors except `$1=|\{\}|={}$ empty
%       argument' in the nominator or denominator, e.g.~$\frac{a+b}{b+a}$ is
%       now represented as $\frac{a+b}1\frac1{b+a}$.}
%    \begin{macrocode}
           \pld@if \pld@Extend\pld@temp{\expandafter\pld@F\pld@frac}\fi
           \expandafter\pld@CF@loop@\pld@symbols\relax\@empty
           \expandafter\pld@CF@loop@\pld@vars\relax\@empty
%    \end{macrocode}
% Finally we add the result to |\pld@tempoly|. Note that |\endgroup| destroys
% all temporary garbage, for example the exponents of symbols and variables.
%    \begin{macrocode}
           \ifx\@empty\pld@temp
               \def\pld@temp{\pld@R11}%
           \fi}%
          \global\let\@gtempa\pld@temp
        \endgroup
        \ifx\@empty\@gtempa\else
            \pld@ExtendPoly\pld@tempoly\@gtempa
        \fi
        \expandafter\pld@CF@loop
    \fi}
%    \end{macrocode}
% For each symbol or variable, we look up the exponent |\pld@@|\meta{symbol}
% and add it together with |\pld@|\meta{\textup{\texttt{S}$\vert$\texttt{V}}}
% and the symbol to the summand |\pld@temp|.
%    \begin{macrocode}
\def\pld@CF@loop@#1#2{%
    \ifx\relax#1\else
        \xdef\@gtempa{\csname pld@@\string#2\endcsname}%
        \ifnum \@gtempa=\z@ \else
            \pld@AddTo\pld@temp{#1{#2}}%
            \pld@Extend\pld@temp{\expandafter{\@gtempa}}%
        \fi
        \expandafter\pld@CF@loop@
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CFSymbol}
% \begin{macro}{\pld@CFVar}
% These definitions collect the exponents. Here we only insert type arguments.
%    \begin{macrocode}
\def\pld@CFSymbol{\pld@CFSV\pld@symbols\pld@S}
\def\pld@CFVar{\pld@CFSV\pld@vars\pld@V}
%    \end{macrocode}
% A new symbol initializes |\pld@@|\meta{symbol} to the current exponent and
% adds the symbol to the list, whereas \ldots
%    \begin{macrocode}
\def\pld@CFSV#1#2#3#4{%
    \@ifundefined{pld@@\string#3}%
    {\@namedef{pld@@\string#3}{#4}%
     \pld@AddTo#1{#2{#3}}}%
%    \end{macrocode}
% an existing one increases |\pld@@|\meta{symbol}.
%    \begin{macrocode}
    {\@tempcnta\csname pld@@\string#3\endcsname\relax
     \advance\@tempcnta#4\relax
     \expandafter\edef\csname pld@@\string#3\endcsname{\the\@tempcnta}}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@CondenseFrac}
% For a fraction, we work on the nominator and denominator separately: a sum
% is added to |\pld@frac|, otherwise we `execute' the code---but of course the
% reciprocal of the denominator. This adds the appropriate data.
%    \begin{macrocode}
\def\pld@CondenseFrac#1#2{%
    \pld@IfSum{#1}{\pld@CFFracAdd{\pld@F{#1}{}}{}}%
                  {#1}%
    \pld@IfSum{#2}{\pld@CFFracAdd{}{\pld@F{}{#2}}}%
                  {\begingroup
                     \pld@DefInverse\pld@temp{#2}%
                     \global\let\@gtempa\pld@temp
                   \endgroup
                   \@gtempa}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CFFracAdd}
% We just add the nominator and denominator to |\pld@frac| and indicate this
% by setting |\pld@if| true.
%    \begin{macrocode}
\def\pld@CFFracAdd{\pld@true \expandafter\pld@CFFracAdd@\pld@frac}
\def\pld@CFFracAdd@#1#2#3#4{\def\pld@frac{{#1#3}{#2#4}}}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Phase III: Condense monomials of same type}
%
% \begin{macro}{\pld@CondenseMonomials}
% $\langle$|\pld@true|$\vert$|pld@false|$\rangle$\meta{polynomial macro}
% \begin{describe}
% This definition sums up the monomials of \meta{polynomial macro}.
% For example, the representation of
% {\makeatletter\pld@Scan{X^2+e^{-1}X^2}^^A
%         $\pld@PrintPoly\pld@tempoly$
% becomes  \pld@CondenseMonomials\pld@true\pld@tempoly
%         $\pld@PrintPoly\pld@tempoly$}.
% If and only if the first argument is |\pld@false|, the macro works on
% symbols instead of variables, for example
% {\makeatletter\pld@Scan{1-\pi+2\pi+e^{-1}+e^{-1}}^^A
%               \pld@CondenseFactors\pld@tempoly
%         $\pld@PrintPoly\pld@tempoly$
% becomes  \pld@CondenseMonomials\pld@false\pld@tempoly
%         $\pld@PrintPoly\pld@tempoly$}.
% \end{describe}
% Again we redefine |\pld@R|,\ldots,|\pld@V|. Here they will add their
% arguments to the current summand. To condense a sum of constants, i.e.~to
% work on symbols, we need to redefine two more macros and sort the constants
% first. To understand this, read ahead and notice the paragraph  at the end
% of this subsection.
%    \begin{macrocode}
\def\pld@CondenseMonomials#1#2{%
    \ifx\@empty#2\else
        \begingroup
          #1%
          \pld@if\else
              \let\pld@SortVars@V\pld@SortVars@S
              \let\pld@SplitMonom\pld@SplitMonomS
              \pld@SortMonomials#2%
          \fi
          \let\pld@R\pld@CMRational
          \let\pld@F\pld@CMFrac
          \let\pld@S\pld@CMSymbol
          \let\pld@V\pld@CMError
%    \end{macrocode}
% Initialize temporary macros, expand the polynomial and work on it, and
% assign the result back to |#1|.
%    \begin{macrocode}
          \let\pld@temp#2\let\pld@tempoly\@empty
          \pld@AccuSetX01\let\pld@symbols\@empty \let\pld@monom\relax
          \expandafter\pld@CM@\pld@temp+\relax+%
          \global\let\@gtempa\pld@tempoly
        \endgroup
        \let#2\@gtempa
    \fi}
%    \end{macrocode}
% Reaching the end of the polynomial, we just add the last summand to the
% temporary polynomial. Otherwise the monomial is split into `factors' and
% `variables', which are handled by |\pld@CM@do|. Afterwards we proceed to
% the next summand.
%    \begin{macrocode}
\def\pld@CM@#1+{%
    \ifx\relax#1\relax
        \pld@CMAddToTempoly
    \else
        \pld@SplitMonom\pld@CM@do{#1}%
        \expandafter\pld@CM@
    \fi}
%    \end{macrocode}
% The following macro gets the nonvariable and variable part as arguments. If
% we haven't worked on a summand yet, we don't need to do anything special. At
% the end of the macro we will add the nonvariable part to the currently empty
% nonvariable part.
%    \begin{macrocode}
\def\pld@CM@do#1#2{%
    \ifx\pld@monom\relax \else
%    \end{macrocode}
% Otherwise we check whether the last and current monomials are of same type.
% Note that this simple |\ifx| requires the variables being in the same order.^^A
% \footnote{It's better to use |\bslash pld@IfMonomE|, but even this requires
%       the mentioned restriction.}
% If the monomials are different, we add the last monomial to the temporary
% polynomial and initialize some macros again.
%    \begin{macrocode}
        \def\pld@temp{#2}%
        \ifx\pld@temp\pld@monom \else
            \pld@CMAddToTempoly
            \pld@AccuSetX01\let\pld@symbols\@empty \let\pld@monom\relax
        \fi
    \fi
%    \end{macrocode}
% In any case we add the nonvariable part to the current (possibly cleared)
% one.
%    \begin{macrocode}
    \let\pld@op+%
    \ifx\@empty#1\@empty \pld@R11\relax \else #1\relax \fi
    \def\pld@monom{#2}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CMAddToTempoly}
% According to whether the accumulator is zero, we save the value in
% |\pld@temp| or just empty that macro.
%    \begin{macrocode}
\def\pld@CMAddToTempoly{%
    \pld@AccuIfZero{\let\pld@temp\@empty}%
                   {\pld@AccuGet\pld@temp
                    \edef\pld@temp{\noexpand\pld@R\pld@temp}}%
%    \end{macrocode}
% Then we we simplify the (possible) sum of symbols by calling the main
% definition of this section with |\pld@false|. Afterwards we append the
% symbols if necessary.
%    \begin{macrocode}
    \pld@CondenseMonomials\pld@false\pld@symbols
    \ifx\pld@symbols\@empty \else
        \pld@ExtendPoly\pld@temp\pld@symbols
    \fi
%    \end{macrocode}
% Now depending on the contents of |\pld@temp|, we do nothing---the sum of the
% monomials is zero---or add the factors together with the `variable part' to
% the polynomial. Note that we put |\pld@F{| and |}{}| around a sum if and
% only if |\pld@if| is true.\footnote{Why we don't need a fraction and also
% don't want it in the other case? We don't want it since it would make things
% more complex. We don't need it since, if we strip off both variables and
% symbols, there are only rationals left and these are evaluated completely
% and condensed in one single rational---no sum, no need for a fraction.}
%    \begin{macrocode}
    \ifx\pld@temp\@empty \else
        \pld@if
            \expandafter\pld@IfSum\expandafter{\pld@temp}%
                {\expandafter\def\expandafter\pld@temp\expandafter
                    {\expandafter\pld@F\expandafter{\pld@temp}{}}}%
                {}%
        \fi
        \pld@ExtendPoly\pld@tempoly\pld@temp
        \pld@Extend\pld@tempoly{\pld@monom}%
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CMFracAdd}
% Depending on the current operator |\pld@op|, we add a new summand to
% |\pld@symbols| or extend the last summand by another factor.
%    \begin{macrocode}
\def\pld@CMFracAdd{%
    \ifx +\pld@op \let\pld@op\@empty
                  \expandafter\pld@AddToPoly
            \else \expandafter\pld@AddTo \fi
    \pld@symbols}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CMRational}
% We add the rational to the accumulator if and only if there is no other
% (symbolic or fractional) factor. This is the reason for some |\relax|es
% above and below.
%    \begin{macrocode}
\def\pld@CMRational#1#2#3{%
    \ifx\relax#3%
        \pld@AccuAdd{#1}{#2}%
    \else
%    \end{macrocode}
% If the rational belongs to a more complex factor, we add it to
% |\pld@symbols|. Note that the used macro was redefined above.
%    \begin{macrocode}
        \pld@CMFracAdd{\pld@R{#1}{#2}}%
        \expandafter#3%
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CMSymbol}
% A symbol is just copied.
%    \begin{macrocode}
\def\pld@CMSymbol#1#2{\pld@CMFracAdd{\pld@S{#1}{#2}}}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CMFrac}
% Here we remove a possibly inserted |\pld@F{| and |}{}| around a sum and
% execute the nominator---or we add the (condensed) fraction.
%    \begin{macrocode}
\def\pld@CMFrac#1#2{%
    \ifx\@empty#2\@empty
        \pld@CMFrac@nom#1+\relax+%
    \else
        \pld@CMFrac@{#1}{#2}%
    \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@CMFrac@#1#2{%
    \pld@IfSum{#1}{\pld@CMFracAdd{\pld@F{#1}{}}}%
                  {#1}%
    \pld@IfSum{#2}{\pld@CMFracAdd{\pld@F{}{#2}}}%
                  {\begingroup
                     \pld@DefInverse\pld@temp{#2}%
                     \global\let\@gtempa\pld@temp
                   \endgroup
                   \@gtempa}}
%    \end{macrocode}
% A nominator is executed summand by summand.
%    \begin{macrocode}
\def\pld@CMFrac@nom#1+{%
    \ifx\relax#1\else
        #1\relax
        \expandafter\pld@CMFrac@nom
    \fi}
%    \end{macrocode}
% \end{macro}
%
% In this section we have made two assumptions: (a) \emph{variables have
% always the same order in monomials} and (b) \emph{monomials of the same
% type}---that means at most different in the nonvariable part---\emph{must
% follow each other}. The first condition has already been mentioned, the
% second is necessary for looking only at the next monomial to check whether
% we have to summarize their preceeding factors. Both are established in the
% following section.
%
%
% \subsection{Phase II: Sort monomials by type}
%
% \begin{macro}{\pld@SortMonomials}
% We first sort the variables of each monomial and then the monomials.
%    \begin{macrocode}
\def\pld@SortMonomials#1{%
    \ifx #1\@empty \else
        \begingroup
%    \end{macrocode}
%    \begin{macrocode}
          \let\pld@temp#1\let\pld@tempoly\@empty
          \expandafter\pld@SortVars\pld@temp+\relax+%
%    \end{macrocode}
%    \begin{macrocode}
          \let\pld@temp\pld@tempoly \let\pld@tempoly\@empty
          \expandafter\pld@SortSummands\pld@temp+\relax+%
%    \end{macrocode}
%    \begin{macrocode}
          \global\let\@gtempa\pld@tempoly
        \endgroup
        \let#1\@gtempa
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@SortVars}
% While not reaching the end \ldots
%    \begin{macrocode}
\def\pld@SortVars#1+{%
    \ifx\relax#1\relax\else
        \pld@SplitMonom\pld@SortVars@{#1}%
        \expandafter\pld@SortVars
    \fi}
%    \end{macrocode}
% we sort the variables if necessary---note the redefinitions of |\pld@V| and
% |\pld@S|---
%    \begin{macrocode}
\def\pld@SortVars@#1#2{%
    \begingroup
      \def\pld@monom{#2}%
      \ifx\@empty\pld@monom\else
          \let\pld@V\pld@SVVar
          \let\pld@S\pld@SVSymbol
          \pld@SortVars@V
      \fi
      \global\let\@gtempa\pld@monom
    \endgroup
%    \end{macrocode}
% and put the things together again.
%    \begin{macrocode}
    \def\pld@factor{#1}%
    \pld@Extend\pld@factor\@gtempa
    \pld@ExtendPoly\pld@tempoly\pld@factor}
%    \end{macrocode}
% We use good old bubble sort on the contents of |\pld@monom|. This macro
% terminates if no items have been interchanged.
%    \begin{macrocode}
\def\pld@SortVars@V{%
    \pld@false \let\pld@temp\pld@monom \let\pld@monom\@empty
    \pld@temp\pld@V\relax\relax
    \pld@if \expandafter\pld@SortVars@V \fi}
%    \end{macrocode}
% The redefinition of |\pld@V| first checks whether the end of the variables
% has been reached. If this is not the case, we either add the current
% variable to |\pld@monom| and continue with the next one, or we interchange
% the variables and indicate this by |\pld@true|.
%    \begin{macrocode}
\def\pld@SVVar#1#2\pld@V#3#4{%
    \ifx\relax#3\relax
        \pld@AddTo\pld@monom{\pld@V{#1}{#2}}%
    \else
        \pld@IfVarL{#1}{#3}{\pld@AddTo\pld@monom{\pld@V{#1}{#2}}%
                            \def\pld@next{\pld@V{#3}{#4}}}%
                           {\pld@true
                            \pld@AddTo\pld@monom{\pld@V{#3}{#4}}%
                            \def\pld@next{\pld@V{#1}{#2}}}%
        \expandafter\pld@next
    \fi}
%    \end{macrocode}
% The similar for |\pld@S| instead of |\pld@V| for sorting symbols.
%    \begin{macrocode}
\def\pld@SortVars@S{%
    \pld@false \let\pld@temp\pld@monom \let\pld@monom\@empty
    \pld@temp\pld@S\relax\relax
    \pld@if \expandafter\pld@SortVars@S \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@SVSymbol#1#2\pld@S#3#4{%
    \ifx\relax#3\relax
        \pld@AddTo\pld@monom{\pld@S{#1}{#2}}%
    \else
        \pld@IfVarL{#1}{#3}{\pld@AddTo\pld@monom{\pld@S{#1}{#2}}%
                            \def\pld@next{\pld@S{#3}{#4}}}%
                           {\pld@true
                            \pld@AddTo\pld@monom{\pld@S{#3}{#4}}%
                            \def\pld@next{\pld@S{#1}{#2}}}%
        \expandafter\pld@next
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@SortSummands}
% Now comes the same for whole monomials except that we can't redefine any
% kind of |\pld@V| and that we will use insertion sort. So, while not reaching
% the end \ldots
%    \begin{macrocode}
\def\pld@SortSummands#1+{%
    \ifx\relax#1\relax\else
%    \end{macrocode}
% we find the right place for |\pld@monom| in |\pld@tempoly|.
%    \begin{macrocode}
        \ifx\@empty\pld@tempoly
            \def\pld@tempoly{#1}%
        \else
            \def\pld@monom{#1}%
            \let\pld@temp\pld@tempoly \let\pld@tempoly\@empty
            \expandafter\pld@SortSummands@i\pld@temp+\relax+%
        \fi
        \expandafter\pld@SortSummands
    \fi}
%    \end{macrocode}
% For this, we iterate down the intermediate result and \ldots
%    \begin{macrocode}
\def\pld@SortSummands@i#1+{%
    \ifx\relax#1\relax
        \pld@ExtendPoly\pld@tempoly\pld@monom
        \expandafter\@gobble
    \else
        \expandafter\pld@SortSummands@j
    \fi
    {#1}}
%    \end{macrocode}
% we test whether we've found the right place and insert the monomial if
% necessary.
%    \begin{macrocode}
\def\pld@SortSummands@j#1{%
    \expandafter\pld@IfMonomL\expandafter{\pld@monom}{#1}%
        {\pld@AddToPoly\pld@tempoly{#1}%
         \pld@SortSummands@i}%
        {\pld@SortSummands@k\pld@monom+#1+}}
\def\pld@SortSummands@k#1+\relax+{\pld@ExtendPoly\pld@tempoly{#1}}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Phase II: Lexicographical order}
%
% \begin{macro}{\pld@IfVarL}
% \marg{varibale 1}\marg{variable 2}\marg{then}\marg{else}
% \begin{describe}
% This macro executes \meta{then} if and only if \meta{variable 1} is less
% than \meta{variable 2} (with respect to the lexicographical order defined
% by this macro).
% \end{describe}
% First we check whether the variables are equal.
%    \begin{macrocode}
\def\pld@IfVarL#1#2{%
    \begingroup
      \def\pld@va{#1}\def\pld@vb{#2}%
      \ifx\pld@va\pld@vb
          \aftergroup\@secondoftwo
      \else
%    \end{macrocode}
% If the variables are not equal, we use their `|\meaning| expansion' for a
% string comparison.
%    \begin{macrocode}
          \edef\pld@next{\expandafter\strip@prefix\meaning\pld@va
                                           \relax\noexpand\@empty
                         \expandafter\strip@prefix\meaning\pld@vb
                                           \relax\noexpand\@empty}%
          \expandafter\pld@IfVarL@\pld@next
      \fi
    \endgroup}
%    \end{macrocode}
% If we've reached the end of a variable, we call the appropriate macro
% |\aftergroup|.
%    \begin{macrocode}
\def\pld@IfVarL@#1#2\@empty#3#4\@empty{%
    \let\pld@next\@empty
          \ifx #3\relax \aftergroup\@secondoftwo
    \else \ifx #1\relax \aftergroup\@firstoftwo
    \else
%    \end{macrocode}
% Otherwise we either need to look at the next characters or compare the two
% ones.
%    \begin{macrocode}
        \ifx#1#3%
            \def\pld@next{\pld@IfVarL#2\@empty#4\@empty}%
        \else
            \ifnum`#1<`#3\relax \aftergroup\@firstoftwo
                          \else \aftergroup\@secondoftwo \fi
        \fi
    \fi \fi
    \pld@next}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@IfMonomL}
% \marg{monomial 1}\marg{monomial 2}\marg{then}\marg{else}
% \begin{describe}
% This macro executes \meta{then} if and only if \meta{monomial 1} is less
% than \meta{monomial 2}. It is required that both monomials' variables are
% sorted.
% \end{describe}
% The implementation is straight forward if you know the last definitions.
%    \begin{macrocode}
\def\pld@IfMonomL#1#2{%
    \begingroup
      \pld@IfMonomL@#1\pld@V\relax\relax\@empty
                    #2\pld@V\relax\relax\@empty
    \endgroup}
%    \end{macrocode}
% If we've reached the end of the variables, we call the appropriate macro.
%    \begin{macrocode}
\def\pld@IfMonomL@#1\pld@V#2#3#4\@empty#5\pld@V#6#7#8\@empty{%
    \let\pld@next\@empty
          \ifx #6\relax \aftergroup\@secondoftwo
    \else \ifx #2\relax \aftergroup\@firstoftwo
    \else
%    \end{macrocode}
% If we have two variables, there are two main cases in which the first
% monomial is smaller: the variable of the first one is smaller or the
% variables are equal but the first exponent is smaller. If both variable and
% exponent match, we have test the next variables.
%    \begin{macrocode}
        \def\pld@va{#2}\def\pld@vb{#6}%
        \ifx\pld@va\pld@vb
            \ifnum#3=#7\relax
                \def\pld@next{\pld@IfMonomL@#4\@empty#8\@empty}%
            \else
                \ifnum#3<#7\relax \aftergroup\@firstoftwo
                            \else \aftergroup\@secondoftwo \fi
            \fi
        \else
            \pld@IfVarL#2\relax\@empty#6\relax\@empty
                {\aftergroup\@firstoftwo}%
                {\aftergroup\@secondoftwo}%
        \fi
    \fi \fi
    \pld@next}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@IfMonomE}
% \marg{monomial 1}\marg{monomial 2}\marg{then}\marg{else}
% \begin{describe}
% This macro executes \meta{then} if and only if \meta{monomial 1} has the
% same variables with identical exponents as \meta{monomial 2}. It is required
% that both monomials' variables are sorted.
% \end{describe}
% We just extract the `variable parts' and compare them with |\ifx|.
%    \begin{macrocode}
\def\pld@IfMonomE#1#2{\pld@IfMonomE@#1\pld@V\@empty#2\pld@V\@empty}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@IfMonomE@#1\pld@V#2\@empty#3\pld@V#4\@empty{%
    \begingroup
      \def\pld@va{#2}\def\pld@vb{#4}%
      \ifx\pld@va\pld@vb \aftergroup\@firstoftwo
                   \else \aftergroup\@secondoftwo \fi
    \endgroup}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Putting together the ingredients}
%
% \begin{macro}{\pld@Simplify}
% \meta{polynomial macro}
% \begin{describe}
% just calls the definitions of the last sections in the correct order.
% \end{describe}
%    \begin{macrocode}
\def\pld@Simplify#1{%
    \pld@CondenseFactors#1%
    \pld@SortMonomials#1%
    \pld@CondenseMonomials\pld@true#1}
%    \end{macrocode}
% \end{macro}
%
%
% \section{Multiplication}
%
% \begin{macro}{\pld@MultiplyPoly}
% \begin{macro}{\pld@NMultiplyPoly}
% \meta{macro a}\meta{macro b}\meta{macro c}
% \begin{describe}
% \meta{macro a} gets \meta{macro b}${}\cdot{}$\meta{macro c} respectively
% the negative polynomial in the second case \texttt{N}.
% \end{describe}
% We use a switch to distinguish the positive and negative form.
%    \begin{macrocode}
\def\pld@MultiplyPoly{\begingroup\pld@true \pld@MultiplyPoly@}
\def\pld@NMultiplyPoly{\begingroup\pld@false \pld@MultiplyPoly@}
%    \end{macrocode}
% Multiply the polynomials and condense the result. Note that the latter is
% the main working procedure. To avoid problems with |\polyhornerscheme| (it
% uses empty macros for the representation of the number 0), we check for
% empty macros here---thanks go to Ludger Humbert.
%    \begin{macrocode}
\def\pld@MultiplyPoly@#1#2#3{%
      \let\pld@temp\@empty
      \ifx\@empty#2\@empty\else \ifx\@empty#3\@empty\else
          \expandafter\pld@MultiplyPoly@a\expandafter#2#3+\relax+%
      \fi \fi
      \global\let\@gtempa\pld@temp
    \endgroup
    \let#1\@gtempa \pld@CondenseFactors#1}
%    \end{macrocode}
% Here we combine each (negated) summand |#2| of the second polynomial with
% \ldots
%    \begin{macrocode}
\def\pld@MultiplyPoly@a#1#2+{%
    \ifx\relax#2\else
        \pld@if \def\pld@va{#2}\else \def\pld@va{#2\pld@R{-1}1}\fi
        \expandafter\pld@MultiplyPoly@b#1+\relax+%
        \expandafter\pld@MultiplyPoly@a\expandafter#1%
    \fi}
%    \end{macrocode}
% each summand |#1| of the first one.
%    \begin{macrocode}
\def\pld@MultiplyPoly@b#1+{%
    \ifx\relax#1\else
        \pld@ExtendPoly\pld@temp{\pld@va#1}%
        \expandafter\pld@MultiplyPoly@b
    \fi}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
%
% \section{Division}
%
%
% \subsection{The algorithm}
%
% \begin{macro}{\pld@DividePoly}
% \begin{macro}{\pld@LongDividePoly}
% A polynomial long division is indicated by |\pld@true|. In this case we also
% need to initialize some macros.
%    \begin{macrocode}
\def\pld@DividePoly{\pld@false \pld@DivPoly}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@LongDividePoly#1#2{%
    \let\pld@pattern\@empty \let\pld@lastline\@empty
    \let\pld@subline\@empty \let\pld@currentline\@empty
    \let\pld@allines\@empty \let\pld@maxcol\z@
    \let\pld@maxcolplustwo\z@ \let\pld@linepos\z@
    \pld@true \pld@DivPoly#1#2%
    \pld@ArrangeResult#1}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% The division algorithm has three main components: the division of two
% monomials, the subtraction of two polynomials, and a loop putting together
% both things.
%
% \begin{macro}{\pld@DivPoly}
% The loop: We initialize remainder, divisor, and quotient.
%    \begin{macrocode}
\def\pld@DivPoly#1#2{%
    \pld@currstage\pld@stage\relax
    \let\pld@remainder#1\let\pld@divisor#2\let\pld@quotient\@empty
    \pld@DivPoly@l}
%    \end{macrocode}
% While the remainder isn't zero and needs to be divided, we extend the
% quotient and subtract the appropriate polynomial from the remainder.
%    \begin{macrocode}
\def\pld@DivPoly@l{%
    \ifx\pld@remainder\@empty\else
        \pld@IfNeedsDivision\pld@remainder\pld@divisor
        {\pld@ExtendPoly\pld@quotient\pld@factor
         \pld@NMultiplyPoly\pld@sub\pld@divisor\pld@factor
         \pld@SubtractPoly\pld@remainder\pld@sub
         \expandafter\pld@DivPoly@l}%
        {\ifpld@InsidePolylongdiv\expandafter\pld@insert@remainder
         \pld@last@remainder+\relax\relax\fi}%
    \fi}
%    \end{macrocode}
% The fix by Hendrik Vogt.
%    \begin{macrocode}
\def\pld@insert@remainder#1+#2\relax{%
    \ifx\relax#1\relax\else\pld@InsertItems\@empty\@empty{#1}\fi
    \ifx\relax#2\relax\else\pld@insert@remainder#2\relax\fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@IfNeedsDivision}
% \meta{polynomial 1}\meta{polynomial 2}\marg{then}\marg{else}
% \begin{describe}
% executes \meta{then} if and only if \meta{polynomial 1} must be divided by
% \meta{polynomial 2} (with possibly nonzero remainder!). |\pld@factor| can be
% used in \meta{then} and holds the quotient of the first monomials. Note that
% this macro requires the polynomials to be sorted.
% \end{describe}
% We expand the two polynomials and add terminators |+\@empty|.
%    \begin{macrocode}
\def\pld@IfNeedsDivision#1#2{%
    \pld@ExpandTwo\pld@IfND{#1+\@empty}{#2+\@empty}}
%    \end{macrocode}
% Now we can divide the first summands of the two polynomials and \ldots
%    \begin{macrocode}
\def\pld@IfND#1+#2\@empty#3+#4\@empty{%
    \pld@DefInverse\pld@factor{#3}%
    \pld@AddTo\pld@factor{#1}%
    \pld@CondenseFactors\pld@factor
%    \end{macrocode}
% check whether all variables have a non-negative exponent.
% Depending on that, we choose the correct argument.
%    \begin{macrocode}
    \begingroup
      \pld@true
      \expandafter\pld@IfND@\pld@factor\pld@V\relax\z@
      \pld@if \aftergroup\@firstoftwo
        \else \aftergroup\@secondoftwo \fi
    \endgroup}
%    \end{macrocode}
% And here we check for (non-)negative exponents.
%    \begin{macrocode}
\def\pld@IfND@#1\pld@V#2#3{%
    \ifx\relax#2\@empty \expandafter\@gobble
                  \else \ifnum#3<\z@ \pld@false \fi
    \fi \pld@IfND@}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@SubtractPoly}
% Here we perform the subtraction. We could define one very short macro for
% `short' and another for polynomial long division, but this macro covers both
% cases. We do nothing if |#2| is empty. This test shouldn't be necessary, but
% who knows how I'll change |\pld@DivPoly| in future.
%    \begin{macrocode}
\def\pld@SubtractPoly#1#2{%
    \ifx#2\@empty\else
%    \end{macrocode}
% For long division, we initialize the horizontal rule's first and last column.
%    \begin{macrocode}
        \pld@if
            \let\pld@firstcol\maxdimen \let\pld@lastcol\z@
        \fi
%    \end{macrocode}
% The submacro does the subtraction and defines appropriate data
% |\pld@lastline|, |\pld@subline|, \ldots\space.
%    \begin{macrocode}
        \let\pld@tempoly\@empty
        \pld@ExpandTwo\pld@SubtractPoly@
                 {#1+\relax+\@empty}{#2+\relax+\@empty}%
        \let#1\pld@tempoly
%    \end{macrocode}
% For long divisions, we now add the calculated lines and a horizontal rule
% to |\pld@allines| (if the current stage allows it). Eventually we reset data.
%    \begin{macrocode}
        \pld@if
            \ifnum\pld@currstage>\z@
                \pld@Extend\pld@allines{\pld@lastline\cr}%
            \else
                \pld@InsertFake\pld@lastline
            \fi
            \advance\pld@currstage-\tw@
            \ifnum\pld@currstage>\z@
                \pld@Extend\pld@allines{\pld@subline\cr}%
                \edef\pld@subline{%
                    \noexpand\cline{\pld@firstcol-\pld@lastcol}
                    \noalign{\vskip\jot}}%
                \pld@Extend\pld@allines\pld@subline
            \else
                \pld@InsertFake\pld@subline
            \fi
            \advance\pld@currstage\m@ne
            \let\pld@lastline\pld@currentline
            \let\pld@subline\@empty
            \let\pld@currentline\@empty
        \fi
    \fi}
%    \end{macrocode}
% The following submacro reads both first monomials. The difference must be
% zero, so we just gobble the monomials except for long division. This is
% coded explicitly to always reduce the degree---no matter what the
% calculations in behind say is true. For long division, we take the
% original monomial, negate it, and use these two for the visualization.
%    \begin{macrocode}
\def\pld@SubtractPoly@#1+#2\@empty#3+{%
    \pld@if
        \pld@DefNegative\pld@monom{#1}%
        \expandafter\pld@InsertItems\expandafter\@empty
            \expandafter{\pld@monom}{#1}%
    \fi
    \pld@SubtractPoly@l#2\@empty}
%    \end{macrocode}
% All other monomials are read here. We have to distinguish several cases.
% If we've reached the end of both polynomials, the next operation is empty.
%    \begin{macrocode}
\def\pld@SubtractPoly@l#1+#2\@empty#3+#4\@empty{%
    \ifx\relax#1\relax
        \let\pld@last@remainder\@empty
        \ifx\relax#3\relax \let\pld@next\@empty \else
%    \end{macrocode}
% If we've reached the end of the first polynomial, we add the monomial of the
% second polynomial, subtract it from the result, use it for visualization, and
% call this macro again to read the rest of the polynomial.
%    \begin{macrocode}
          \pld@AddToPoly\pld@tempoly{#3}%
          \pld@if \pld@InsertItems{#3}{#3}{}\fi
          \def\pld@next{\pld@SubtractPoly@l\relax+\@empty#4\@empty}%
        \fi
    \else
    \ifx\relax#3\relax
%    \end{macrocode}
% If we've reached the end of the second polynomial, we just add the rest of
% the first polynomial to the result |\pld@tempoly|.
%    \begin{macrocode}
        \pld@SubtractPoly@r#1+#2\@empty
        \let\pld@next\@empty
    \else
%    \end{macrocode}
% There are three cases if we have two monomials. If they are equal---which
% means that the variables and their exponents match---, we add the monomials
% and use the result to extend the temporary polynomial as well as for the
% visualization.
%    \begin{macrocode}
        \pld@IfMonomE{#1}{#3}%
        {\def\pld@temp{#1+#3}%
         \pld@CondenseMonomials\pld@true\pld@temp
         \ifx\pld@temp\@empty\else
             \pld@ExtendPoly\pld@tempoly\pld@temp
         \fi
         \pld@if \expandafter\pld@InsertItems\expandafter
                 {\pld@temp}{#3}{#1}\fi
         \def\pld@next{\pld@SubtractPoly@l#2\@empty#4\@empty}}%
%    \end{macrocode}
% If the second monomial is stricly smaller, we extend the temporary polynomial
% and the visualization by this monomial and re-insert the first monomial to be
% read again.
%    \begin{macrocode}
        {\pld@IfMonomL{#1}{#3}%
         {\pld@AddToPoly\pld@tempoly{#3}%
          \pld@if \pld@InsertItems{#3}{#3}{}\fi
          \def\pld@next{\pld@SubtractPoly@l#1+#2\@empty#4\@empty}}%
%    \end{macrocode}
% If the first monomial is stricly smaller, we extend the temporary polynomial
% and the visualization by this monomial and re-insert the other to be read
% again (since we haven't reached the correct place in the table yet).
% Note that these two last cases are some kind of insertion sort.
%    \begin{macrocode}
         {\pld@AddToPoly\pld@tempoly{#1}%
          \pld@if \pld@InsertItems{#1}{}{#1}\fi
          \def\pld@next{\pld@SubtractPoly@l#2\@empty#3+#4\@empty}}%
        }%
    \fi \fi
    \pld@next}
%    \end{macrocode}
% Finally the macro used to add the rest of the first polynomial.
%    \begin{macrocode}
\def\pld@SubtractPoly@r#1+\relax+\@empty{%
    \pld@AddToPoly\pld@tempoly{#1}%
    \def\pld@last@remainder{#1}}
%    \end{macrocode}
% \end{macro}
%
%
% \subsection{Tweaking the alignment}
%
% Here comes important code for partial output of long divisions.
%
% \begin{macro}{\pld@InsertFake}
% This macro is somewhat like |\pld@InsertItems| but gets a whole line. Thus
% we iterate down each entry and compare its width with the next dimension
% from |\pld@fakeline| (which is the current width of the column). In detail:
% We iterate down each entry and \ldots
%    \begin{macrocode}
\def\pld@InsertFake#1{%
    \let\pld@temp\@empty
    \expandafter\pld@InsertFake@l#1&\relax&}
%    \end{macrocode}
% \ldots\space either append the rest of |\pld@fakeline| or get the next
% dimension from the macro.
%    \begin{macrocode}
\def\pld@InsertFake@l#1&{%
    \ifx\relax#1\@empty
        \pld@Extend\pld@temp{\expandafter&\pld@fakeline}%
        \let\pld@fakeline\pld@temp
    \else
        \expandafter\pld@InsertFake@do\pld@fakeline\relax{#1}%
        \expandafter\pld@InsertFake@l
    \fi}
\def\pld@InsertFake@do#1&#2\relax#3{%
%    \end{macrocode}
% We assign the remaining column dimensions or, if there is no dimension left,
% we insert 0pt.
%    \begin{macrocode}
    \ifx\@empty#2\@empty \def\pld@fakeline{0pt&}%
                   \else \def\pld@fakeline{#2}\fi
    \@tempdima#1\relax
    \setbox\z@=\hbox{\ensuremath{#3}}%
%    \end{macrocode}
% Then we add the maximum of the current dimension and the width of |#3| to
% |\pld@temp| (which will be assigned to |\pld@fakeline| as seen above).
%    \begin{macrocode}
    \ifdim\@tempdima<\wd\z@ \@tempdima=\wd\z@ \fi
    \ifx\pld@temp\@empty
        \edef\pld@temp{\the\@tempdima}%
    \else
        \pld@Extend\pld@temp{\expandafter&\the\@tempdima}%
    \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@fakeline{0pt&}% init
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@ConvertFake}
% The contents of |\pld@fakeline| are converted into an appropriate sequence
% of |\vrule|\texttt{ height 0pt depth 0pt width }\meta{column dimension}.
% This put inside the |\halign| below will ensure stable column widths.
%    \begin{macrocode}
\def\pld@ConvertFake#1&{%
    \ifx\relax#1\@empty\else
        \ifx\@empty#1\@empty
            &%
        \else
            \noexpand\vrule\noexpand\@height\z@\noexpand\@depth\z@
                           \noexpand\@width#1\relax&%
        \fi
        \expandafter\pld@ConvertFake
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@SplitQuotient}
% splits |\pld@quotient| into the visible |\pld@real| and invisible
% |\pld@shadow|---according to the given |\pld@stage|. We just iterate down
% the summands:
%    \begin{macrocode}
\def\pld@SplitQuotient{%
    \let\pld@real\@empty \let\pld@shadow\empty
    \pld@currstage\pld@stage\relax
    \expandafter\pld@SplitQuotient@\pld@quotient+\relax+}
\def\pld@SplitQuotient@#1+{%
    \ifx\relax#1\@empty
%    \end{macrocode}
% reaching the end, we check whether the remainder needs to be printed;
%    \begin{macrocode}
            \advance\pld@currstage-\tw@
            \ifnum\pld@currstage<\z@
                \let\pld@PrintRemain\pld@XPLD
            \else
                \let\pld@PrintRemain\pld@PLD
            \fi
    \else
%    \end{macrocode}
% otherwise we either add the current summand to |\pld@real| or |\pld@shadow|.
%    \begin{macrocode}
        \ifx\@empty#1\@empty\else
            \advance\pld@currstage-\tw@
            \ifnum\pld@currstage<\z@
                \pld@AddToPoly\pld@shadow{#1}%
            \else
                \pld@AddToPoly\pld@real{#1}%
            \fi
            \advance\pld@currstage\m@ne
        \fi
        \expandafter\pld@SplitQuotient@
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintPolyShadow}
% prints |\pld@real| and leaves space for |\pld@shadow|.
%    \begin{macrocode}
\def\pld@PrintPolyShadow{%
    \pld@firsttrue
    \ifx\pld@real\@empty\else
        \expandafter\pld@PrintMonoms\pld@real+\relax+%
    \fi
    \ifx\pld@shadow\@empty\else
        \setbox\z@\hbox{$\expandafter\pld@PrintMonoms\pld@shadow
                                                     +\relax+$}%
        \phantom{\copy\z@}%
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \subsection{Aligning long division}\label{iAligningLongDivision}
%
% \begin{macro}{\pld@PrintLongDiv}
% does the horizontal alignment. It puts |\pld@allines| into |\halign|.
%    \begin{macrocode}
\def\pld@PrintLongDiv{%
    \ensuremath{\hbox{\vtop{\begingroup
          \offinterlineskip \tabskip=\z@
  \edef\pld@fakeline{\expandafter\pld@ConvertFake\pld@fakeline&\relax&}%
          \halign{\strut\pld@firsttrue\hfil$##$%
                       &\pld@firsttrue\hfil$##$%
                       &&\hfil$##$\cr
                  \pld@fakeline\cr \noalign{\vskip-\normalbaselineskip}%
                  \pld@allines}%
          \endgroup}}}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@InsertItems}
% Here now we place the monomials. |\pld@pattern| gives the columns in which
% monomials have been put and thus has to be put now. So we first define the
% monomial and look for it in |\pld@pattern|.
%    \begin{macrocode}
\def\pld@InsertItems#1#2#3{%
    \ifx\@empty#1\@empty
    \ifx\@empty#2\@empty \def\pld@monom{#3}%
                   \else \def\pld@monom{#2}\fi
                   \else \def\pld@monom{#1}\fi
    \@tempcnta\@ne \let\pld@recentmonom\@empty
    \expandafter\pld@InsertItems@find\pld@pattern\relax&%
%    \end{macrocode}
% This column |\@tempcnta| must not exceed the current column range, which is
% used to draw the horizontal line: we change the range if necessary.
%    \begin{macrocode}
    \ifnum\pld@firstcol>\@tempcnta \edef\pld@firstcol{\the\@tempcnta}\fi
    \ifnum\pld@lastcol<\@tempcnta \edef\pld@lastcol{\the\@tempcnta}\fi
    \ifnum\pld@maxcol<\@tempcnta 
      \edef\pld@maxcol{\the\@tempcnta}
      \@tempcntb\pld@maxcol\relax\advance\@tempcntb\tw@
      \edef\pld@maxcolplustwo{\the\@tempcntb}
    \fi
%    \end{macrocode}
% Finally we insert the arguments.
%    \begin{macrocode}
    \pld@InsertItems@do\pld@lastline{\pld@PLD{#3}}%
    \pld@InsertItems@do\pld@subline{\pld@PLD{#2}}%
    \pld@InsertItems@do\pld@currentline{\pld@PLD{#1}}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@InsertItems@do}
% For this, we iterate down the specified line |#1|---five items at the same
% time---\ldots
%    \begin{macrocode}
\def\pld@InsertItems@do#1#2{%
    \let\pld@temp\@empty \@tempcntb\@tempcnta
    \expandafter\pld@InsertItems@do@a#1&&&&&\relax{#2}%
    \let#1\pld@temp}
%    \end{macrocode}
% until we've found the item number |\@tempcnta|$=$|\@tempcntb|$-k\cdot 5$.
%    \begin{macrocode}
\def\pld@InsertItems@do@a#1&#2&#3&#4&#5&#6\relax{%
    \ifcase\@tempcntb \or
    \or \pld@AddTo\pld@temp{#1&}%
    \or \pld@AddTo\pld@temp{#1&#2&}%
    \or \pld@AddTo\pld@temp{#1&#2&#3&}%
    \or \pld@AddTo\pld@temp{#1&#2&#3&#4&}%
    \else
%    \end{macrocode}
% If this is not the case, we call this macro again.
%    \begin{macrocode}
        \pld@AddTo\pld@temp{#1&#2&#3&#4&#5&}%
        \advance\@tempcntb-5\relax
        \def\pld@next{\pld@InsertItems@do@a#6&&&&&\relax}%
        \expandafter\@firstoftwo\expandafter\pld@next
    \fi
%    \end{macrocode}
% Otherwise we add the monomial to |\pld@temp|, which is assigned to the
% correct macro above.
%    \begin{macrocode}
    \pld@InsertItems@do@b}
\def\pld@InsertItems@do@b#1{\pld@AddTo\pld@temp{#1}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@InsertItems@find}
% To find the monomial in |\pld@pattern|, we just test each monomial against
% the defined |\pld@monom|. If we've reached the end of the pattern, we `append'
% (see below) the monomial to the pattern and we're done for.
%    \begin{macrocode}
\def\pld@InsertItems@find#1&{%
    \ifx\relax#1\relax
        \expandafter\pld@InsertItems@find@fill\pld@recentmonom\pld@V{}0\@empty
    \else
%    \end{macrocode}
% Otherwise we either drop the rest of the pattern since we've found the
% monomial, or we advance the temporary counter and continue.
%    \begin{macrocode}
        \def\pld@recentmonom{#1}%
        \expandafter\pld@IfMonomE\expandafter{\pld@monom}{#1}%
            {\expandafter\pld@InsertItems@find@\expandafter&}%
            {\advance\@tempcnta\@ne \expandafter\pld@InsertItems@find}%
    \fi}
\def\pld@InsertItems@find@#1&\relax&{}
%    \end{macrocode}
% And now for `appending' a monomial to the pattern. Thanks to Karl Heinz
% Marbaise wrong implementation has been replaced by filling in also the
% monomials between the most recent and current monomial---if that wouldn't be
% done, a higher degree monomial could be preceded by a lower degree one and
% thus would never get printed as $x^7$ in |\polylongdiv{x^{15}+1}{x^5+x^3+x+1}|.
%    \begin{macrocode}
\def\pld@InsertItems@find@fill#1\pld@V#2#3#4\@empty{%
    \expandafter\pld@InsertItems@find@fill@\pld@monom\pld@V{}0\@empty{#3}}
\def\pld@InsertItems@find@fill@#1\pld@V#2#3#4\@empty#5{%
    \ifx\pld@pattern\@empty
        \def\pld@pattern{\pld@V&\pld@V{#2}{#3}&}%
        \@tempcnta\tw@
    \else
        \@tempcntb#5\relax
        \loop \ifnum #3<\@tempcntb
            \advance\@tempcnta\@ne
            \advance\@tempcntb\m@ne
            \ifnum\@tempcntb=\z@
                \def\pld@temp{#1}%
            \else
                \edef\pld@temp{\noexpand\pld@V{#2}{\the\@tempcntb}}%
            \fi
            \pld@Extend\pld@pattern{\pld@temp&}%
        \repeat
        \advance\@tempcnta\m@ne
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@ArrangeResult}
% Here the dividend, divisor, and quotient are added to the `|\halign|' data
% macro. First we add a $0$ below the last horizontal rule if the remainder is
% zero.
%    \begin{macrocode}
\def\pld@ArrangeResult#1{%
    \ifx\pld@remainder\@empty
        \@tempcnta\pld@maxcol\relax
        \pld@InsertItems@do\pld@lastline
            {\pld@firsttrue\pld@PLD{\pld@R{0}{1}}}%
    \fi
    \ifnum\pld@currstage>\z@
        \pld@Extend\pld@allines{\pld@lastline\cr}%
    \else
        \pld@InsertFake\pld@lastline
    \fi
%    \end{macrocode}
% We begin to build the first line. For the quotient printed atop, the divisor
% is the first element. Otherwise we either use a left parentheses or just let
% the first element empty. Note that the size of the parentheses is hard-wired.
%    \begin{macrocode}
    \pld@iftopresult
        \def\pld@lastline{\pld@PrintPoly\pld@divisor\bigr)&}%
    \else
        \let\pld@lastline\@empty
        \ifx C\pld@style
          \def\pld@lastline{\pld@leftdelim\strut\pld@rightxdelim&}%
        \fi
    \fi
%    \end{macrocode}
% Now we put the monomials of the dividend in the correct columns and split the
% quotient into its visible and invisible part.
%    \begin{macrocode}
    \expandafter\pld@AR@col\expandafter\pld@PLD
                           \expandafter\pld@lastline#1+\relax+%
    \pld@SplitQuotient
%    \end{macrocode}
% For a result at top, we put the quotient into |\pld@currentline| and add a
% horizontal line below it.
%    \begin{macrocode}
    \pld@iftopresult
        \let\pld@currentline\@empty
        \expandafter\pld@AR@col\expandafter\pld@PLD
                               \expandafter\pld@currentline
                                           \pld@quotient+\relax+%
        \expandafter\pld@AR@col\expandafter\pld@XPLD
                               \expandafter\pld@currentline
                                           \pld@shadow+\relax+%
        \edef\pld@subline{%
            \noexpand\cline{\tw@-\pld@maxcol}%
            \noalign{\vskip\jot}}%
        \pld@Extend\pld@currentline{\expandafter\cr\pld@subline}%
    \else
%    \end{macrocode}
% For a result next to the dividend, we first calculate the number of columns
% it spans. It's the maximal column minus the last column of the dividend
% (which is |\@tempcnta|) plus one extra column not to squeeze it all into
% the last column of the `normal' table.
%    \begin{macrocode}
        \@tempcnta-\@tempcnta
        \advance\@tempcnta\pld@maxcol\relax \advance\@tempcnta\@ne
        \edef\pld@span{\the\@tempcnta}%
        \@tempcntb\pld@maxcol\relax\advance\@tempcntb\pld@span%
        \advance\@tempcntb\@ne%
        \edef\pld@linepos{\the\@tempcntb}%
%    \end{macrocode}
% Then we can add divisor, quotient, and remainder. First we go for style B.
%    \begin{macrocode}
        \ifx B\pld@style
          \pld@AddTo\pld@lastline{%
            &\multispan\pld@span${}=%
            \pld@PrintPolyWithDelims\pld@divisor
            \expandafter\pld@IfSum\expandafter{\pld@divisor}{}{\cdot}%
            \expandafter\pld@IfSum\expandafter{\pld@quotient}\pld@true
                                                             \pld@false
            \pld@if \pld@leftdelim
                    \pld@PrintPolyShadow
                    \pld@rightdelim
              \else \pld@PrintPolyShadow \fi
            \pld@firstfalse
            \expandafter\pld@PrintRemain\expandafter{\pld@remainder}$}%
        \else
%    \end{macrocode}
% And now for style C. Note that we `smash' the depth of the fraction.
%    \begin{macrocode}
          \if C\pld@style
            \pld@AddTo\pld@lastline{%
              &\multispan\pld@span$\pld@leftxdelim\strut\pld@rightdelim
              \pld@div
              \pld@PrintPolyWithDelims\pld@divisor=
              \pld@PrintPolyShadow
              \ifx\pld@remainder\@empty\else
                  +{}%
                  \setbox\z@=\hbox{$\displaystyle
                    \frac{\let\strut\@empty\pld@firsttrue \expandafter
                          \pld@PrintRemain\expandafter{\pld@remainder}}%
                         {\let\strut\@empty\pld@PrintPoly\pld@divisor}$}%
                  \dp\z@=\z@\box\z@
              \fi
              $}%
          \else
%    \end{macrocode}
% Finally, style D.
%    \begin{macrocode}
            \pld@AddTo\pld@lastline{%
              \cr%
              \noalign{\vskip-\normalbaselineskip}%
              \multispan\pld@maxcol~&~&\multispan\pld@span${}\vrule~%
              \pld@PrintPoly\pld@divisor\hfil\hfil$\cr%
              \cline{\pld@maxcolplustwo-\pld@linepos}%
              \multispan\pld@maxcol~&~&\multispan\pld@span${}\vrule height
              2.25ex~\pld@PrintPolyShadow$\hfil\cr\noalign{\vskip-2\normalbaselineskip}}%
          \fi
        \fi
    \fi
%    \end{macrocode}
% Eventually we replace the first line in |\pld@allines| by |\pld@lastline|
% or add |\pld@currentline| before doing so.
%    \begin{macrocode}
    \expandafter\pld@AR@\pld@allines\relax}
\def\pld@AR@#1\cr#2\relax{%
    \pld@iftopresult
        \let\pld@allines\pld@currentline
        \pld@AddTo\pld@allines{\pld@lastline\cr #2}%
    \else
        \let\pld@allines\pld@lastline
        \pld@AddTo\pld@allines{\cr #2}%
    \fi}
%    \end{macrocode}
% The dividend and quotient above are built by looking up the position of each
% monomial in |\pld@pattern| and inserting these monomials. |#1|, which is
% |\pld@PLD| or |\pld@XPLD|, is used to print the monomial.
%    \begin{macrocode}
\def\pld@AR@col#1#2#3+{%
    \ifx\relax#3\@empty\else
        \ifx\@empty#3\@empty\else
            \def\pld@monom{#3}\@tempcnta\@ne
            \expandafter\pld@InsertItems@find\pld@pattern\relax&%
            \pld@InsertItems@do#2{#1{#3}}%
        \fi
        \expandafter\pld@AR@col\expandafter#1\expandafter#2%
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PLD}
% \begin{macro}{\pld@XPLD}
% have been used above several times. They print single monomials in the
% horizontal alignment of a long division or put a |\phantom| around it.
%    \begin{macrocode}
\def\pld@PLD#1{\ifx\@empty#1\@empty\else\pld@PrintMonoms#1+\relax+\fi}
\def\pld@XPLD#1{\phantom{\pld@PLD{#1}}}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
%
% \section{Euclidean algorithm}
%
% \begin{macro}{\pld@LongEuclideanPoly}
% Assign the `smaller' polynom to |\pld@remainder| and the other to
% |\pld@divisor| by doing one or two divisions. Additionally |\pld@vb| is
% initialized for the case that no further division is needed (v0.15).
%    \begin{macrocode}
\def\pld@LongEuclideanPoly#1#2{%
    \pld@false \let\pld@allines\@empty
    \pld@DivPoly#1#2%
    \ifx\pld@quotient\@empty
        \pld@DivPoly#2#1%
        \pld@InsertEuclidean#2#1%
        \let\pld@vb#1%
    \else
        \pld@InsertEuclidean#1#2%
        \let\pld@vb#2%
    \fi
%    \end{macrocode}
% Now we start the well known Euclidean algorithm. |\pld@va| and |\pld@vb|
% are used as temporary scratch `registers'.
%    \begin{macrocode}
    \pld@LongEuclideanPoly@l}
\def\pld@LongEuclideanPoly@l{%
    \ifx\pld@remainder\@empty \else
        \let\pld@va\pld@divisor
        \let\pld@vb\pld@remainder
%    \end{macrocode}
%    \begin{macrocode}
        \pld@DivPoly\pld@va\pld@vb
        \pld@Simplify\pld@quotient \pld@Simplify\pld@remainder
        \pld@InsertEuclidean\pld@va\pld@vb
%    \end{macrocode}
%    \begin{macrocode}
        \expandafter\pld@LongEuclideanPoly@l
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@InsertEuclidean}
% Each step inserts one line with dividend, divisor, quotient, and remainder.
%    \begin{macrocode}
\def\pld@InsertEuclidean#1#2{%
    \ifx \pld@allines\@empty \else
        \pld@AddTo\pld@allines{\noalign{\vskip\jot}}%
    \fi
    \pld@Extend\pld@allines{\expandafter\pld@PrintPolyArg
                            \expandafter{#1}&}%
    \pld@Extend\pld@allines{\expandafter\pld@PrintPolyWithDelimsArg
                            \expandafter{#2}\hfil\cdot\hfil}%
    \pld@Extend\pld@allines{\expandafter\pld@PrintPolyWithDelimsArg
                            \expandafter{\pld@quotient}&}%
    \pld@Extend\pld@allines{\expandafter\pld@PrintPolyWithDelimsArg
                            \expandafter{\pld@remainder}\cr}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@PrintLongEuclidean}
% just like |\pld@PrintLongDiv|.
%    \begin{macrocode}
\def\pld@PrintLongEuclidean{
    \ensuremath{\hbox{\vtop{\begingroup
          \offinterlineskip \tabskip=\z@
          \halign{\strut\pld@firsttrue\hfil$##$%
                  &${}={}$\hfil$##$\hfil
                  &${}+##$\hfil\cr \pld@allines}%
          \endgroup}}}}
%    \end{macrocode}
% \end{macro}
%
%
% \section{Factorization}
%
% The algorithm is based on the following proposition:
% All \emph{rational} zeros of a polynomial $a_nX^n+\ldots+a_1X+a_0$ with
% \emph{integer} coefficients are among the fractions $\pm\frac\beta\alpha$
% where $\beta$ is a divisor of $a_0$ and $\alpha$ a divisor of the leading
% coefficient $a_n$.
% So our first tasks are to iterate through divisors and to test for zeros.
%
% \begin{macro}{\pld@NextDivisorPair}
% \marg{integer $a$}\marg{integer $b$}
% \begin{describe}
% |\@tempcnta| and |\@tempcntb| get the next divisors of \meta{integer $a$}
% and \meta{integer $b$}. |\pld@if| is set false if and only if all divisor
% pairs has been iterated through. At the beginning we must initialize
% |\@tempcnta=\z@| and |\@tempcntb=\@ne|.
% \end{describe}
% First |\@tempcnta| becomes the next divisor of |#1| and then |\@tempcntb|
% the next one of |#2| if and only if |#1| has no more divisors (which resets
% |\@tempcnta| automatically).
%    \begin{macrocode}
\def\pld@NextDivisorPair#1#2{%
    \pld@NextDivisor\@tempcnta{#1}%
    \pld@if\else
        \pld@NextDivisor\@tempcntb{#2}%
    \fi}
%    \end{macrocode}
% Here we advance the counter by one until the counter gets too big (note that
% this `$>$' requires the arguments to |\pld@NextDivisorPair| being positive)
% \ldots
%    \begin{macrocode}
\def\pld@NextDivisor#1#2{%
    \advance#1\@ne
    \ifnum #1>#2\relax
        #1\@ne \pld@false
        \expandafter\@gobbletwo
    \else
%    \end{macrocode}
% or a divisor of |#2|.
%    \begin{macrocode}
        \@multicnt #2\relax
        \divide\@multicnt#1\multiply\@multicnt#1%
        \advance\@multicnt-#2\relax
        \ifnum \@multicnt=\z@
            \pld@true
            \expandafter\expandafter\expandafter\@gobbletwo
        \else
            \expandafter\expandafter\expandafter\pld@NextDivisor
        \fi
    \fi
    #1{#2}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@FindZeros}
% \marg{integer $a$}\marg{integer $b$}
% \begin{describe}
% is the main loop: while not all divisor pairs has been processed, we check
% whether $\pm$|\@tempcntb|$/$|\@tempcnta| is a zero.
% \end{describe}
%    \begin{macrocode}
\def\pld@FindZeros#1#2{%
    \pld@NextDivisorPair{#1}{#2}%
    \pld@if
        \pld@CheckZeros
        \def\pld@next{\pld@FindZeros{#1}{#2}}%
        \expandafter\pld@next
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@CheckZeros}
% When $\frac\beta\alpha$ isn't a zero any more, we add the zero with
% multiplicity |\@multicnt| \ldots
%    \begin{macrocode}
\def\pld@CheckZeros{%
    \pld@true \@multicnt\z@
    \loop \pld@if
        \pld@CheckZero{\the\@tempcnta}{\the\@tempcntb}%
    \repeat
    \pld@AddRationalZero{\the\@tempcnta}{\the\@tempcntb}%
%    \end{macrocode}
% and do the same for $-\frac\beta\alpha$. Note that the multiplicity might be
% zero.
%    \begin{macrocode}
    \pld@true \@multicnt\z@
    \loop \pld@if
        \pld@CheckZero{-\the\@tempcnta}{\the\@tempcntb}%
    \repeat
    \pld@AddRationalZero{-\the\@tempcnta}{\the\@tempcntb}}
%    \end{macrocode}
% To check for the zero $\frac\beta\alpha$, we divide |\pld@current| by the
% linear factor $X-\frac\beta\alpha$. Note that |\pld@tempoly| contains the
% string |\pld@V{X}1| where |X| is replaced by the actual variable; so we just
% need to append the fraction.
%    \begin{macrocode}
\def\pld@CheckZero#1#2{%
    \begingroup
      \edef\pld@temp{{-#2}{#1}}%
      \pld@Extend\pld@tempoly{\expandafter+\expandafter\pld@R\pld@temp}%
      \let\pld@stage\maxdimen \pld@DividePoly\pld@current\pld@tempoly
      \ifx\pld@remainder\@empty
          \global\let\@gtempa\pld@quotient
          \aftergroup\pld@true
      \else
          \aftergroup\pld@false
      \fi
    \endgroup
%    \end{macrocode}
% If the division was successful, we advance the multiplicity and assign the
% new polynomial.
%    \begin{macrocode}
    \pld@if
        \advance\@multicnt\@ne
        \let\pld@current\@gtempa
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AddRationalZero}
% Here we add code to |\pld@allines| to \emph{print} the factor with its
% multiplicity.
%    \begin{macrocode}
\def\pld@AddRationalZero#1#2{%
    \ifnum\@multicnt=\z@\else
        \pld@AccuSetX{#2}{-#1}%
        \pld@AccuGet\pld@temp
        \edef\pld@temp{\noexpand\pld@R\pld@temp}%
%    \end{macrocode}
% Yet |\pld@temp| contains the rational. Note that the `accumulator detour' is
% needed to get rid of |\@tempcnta| and |b|. Eventually append the zero with
% the exponent if necessary.
%    \begin{macrocode}
        \expandafter\pld@AddZero\expandafter{\pld@temp}%
        \ifnum\@multicnt=\@ne\else
            \edef\pld@temp{^{\the\@multicnt}}%
            \pld@Extend\pld@allines\pld@temp
        \fi
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AddZero}
% We add |\pld@leftdelim| |\pld@firsttrue| |\pld@PLD{X+#1}| |\pld@rightdelim|
% to the current factorization |\pld@allines|.
%    \begin{macrocode}
\def\pld@AddZero#1{%
    \pld@Extend\pld@allines{\expandafter\pld@leftdelim
                            \expandafter\pld@firsttrue
                            \expandafter\pld@PLD
                            \expandafter{\pld@tempoly+#1}%
                            \pld@rightdelim}}
%    \end{macrocode}
% \end{macro}
%
% These are the basic definitions. Now remember that the proposition above
% requires integers coefficients, but we want to support rationals. To do
% this, we multiply the polynomial virtually by the least common multiple of
% all denominators. `Virtually' means that we only multiply the leading
% coefficient and the absolute term to get the correct divisors, but not the
% real polynomial.
%
% \begin{macro}{\pld@FactorizeInit}
% And this is done here. The argument is a definition to be executed with
% appropriate data (the multiplied coeffients) as arguments at the end of this
% macro. Again we redefine |\pld@R|,\ldots,|\pld@V| and iterate through the
% monomials. The accumulator holds the least common multiple and |\@multicnt|
% the least exponent of the variable (since we need to divide by $X^k$ to get
% an absolute term).
%    \begin{macrocode}
\def\pld@FactorizeInit#1{%
    \begingroup
      \pld@firsttrue \let\pld@sub\@empty
      \pld@AccuSetX11%
      \let\pld@R\pld@FRational
      \let\pld@F\@gobbletwo
      \let\pld@S\@gobbletwo
      \let\pld@V\pld@FVar
      \expandafter\pld@FactorizeInit@\pld@current+\relax+%
%    \end{macrocode}
% Below you'll see |\@gtemp| $=$ leading coeffients and |\pld@lastline| $=$
% coefficient of absolute term (after division by $X^k$). Here we multiply by
% the accumulator and make the results positive (if we're advised to do this)
% and \ldots
%    \begin{macrocode}
      \pld@if
          \pld@AccuGet\pld@temp
          \expandafter\pld@AccuMul\@gtempa
          \pld@AccuIfNegative{\pld@AccuNegate}{}%
          \pld@AccuGet\pld@va
          \expandafter\pld@AccuSetX\pld@temp
          \expandafter\pld@AccuMul\pld@lastline
          \pld@AccuIfNegative{\pld@AccuNegate}{}%
          \pld@AccuGet\pld@vb
      \else
          \let\pld@va\@gtempa
          \let\pld@vb\pld@lastline
      \fi
%    \end{macrocode}
% set the coefficient of $X^1$ if necessary---or any other variable power 1.
%    \begin{macrocode}
      \ifx\pld@sub\@empty \def\pld@sub{01}\fi
%    \end{macrocode}
% Then we prepare the arguments for the macro to be executed at the end. In
% particular, |\pld@tempoly| is defined to define |\def\pld@tempoly{\pld@V{X}}|
% below, and this definition is cared out before we execute the macro |#2|.
%    \begin{macrocode}
      \edef\pld@temp{\noexpand#1\pld@va\pld@vb{\the\@multicnt}\pld@sub}%
      \pld@Extend\pld@tempoly{\pld@temp}%
      \global\let\@gtempa\pld@tempoly
    \endgroup
    \@gtempa}
%    \end{macrocode}
% The submacro just iterates down the monomials.
%    \begin{macrocode}
\def\pld@FactorizeInit@#1+{%
    \ifx\relax#1\else
        \def\pld@lastline{11}%
        #1%
        \expandafter\pld@FactorizeInit@
    \fi}
%    \end{macrocode}
% The following two definitions store the leading coefficient in |\@gtempa|,
% the last in |\pld@lastline|, the coefficient of $X^1$ in |\pld@sub|, update
% the least common multiple, \ldots
%    \begin{macrocode}
\def\pld@FRational#1#2{%
    \def\pld@lastline{{#1}{#2}}%
    \pld@iffirst
        \global\let\@gtempa\pld@lastline
        \def\pld@tempoly{\@multicnt\z@}%
    \fi
    \pld@LCM{#2}%
    \@multicnt\z@}
%    \end{macrocode}
% and save the variable and its `leading' exponent in |\@multicnt|. Note that
% these two definitions are cared out later on, and the assignment of
% |\@multicnt| here saves the exponent of the last monomial.
%    \begin{macrocode}
\def\pld@FVar#1#2{%
    \pld@iffirst
        \pld@firstfalse
        \global\let\@gtempa\pld@lastline
        \def\pld@tempoly{\def\pld@tempoly{\pld@V{#1}}%
                         \@multicnt#2\relax}%
    \fi
    \@multicnt#2\relax
%    \end{macrocode}
%    \begin{macrocode}
    \ifnum\@multicnt=\@ne
        \let\pld@sub\pld@lastline
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@Factorize}
% If the given polynomial is zero, the factorization is `$0$'. Otherwise we
% initialize data and start the algorithm. Note that |\pld@Factorize@| is an
% argument to |\pld@FactorizeInit| and called from inside with appropriate
% arguments.
%    \begin{macrocode}
\def\pld@Factorize#1{%
    \ifx\@empty#1\@empty
        \def\pld@allines{\pld@PrintPolyWithDelims\@empty}%
    \else
        \let\pld@allines\@empty
        \let\pld@current#1%
        \pld@true \pld@FactorizeInit
        \pld@Factorize@
    \fi}
%    \end{macrocode}
% Now the arguments are
%   |#1#2|$=$\meta{`leading coefficient'},
%   |#3#4|$=$\meta{`coefficient of least monomial'},
%   |#5|$=$\meta{least exponent},
%   |#6#7|$=$\meta{coefficient of linear summand}.
% Here the first two coefficient have been multiplied by the least common
% multiple of all denominators. The factorization gets `variable power |#5|'
% and the current polynomial is divided this.
%    \begin{macrocode}
\def\pld@Factorize@#1#2#3#4#5#6#7{%
    \ifnum #5=\z@\else
        \pld@Extend\pld@allines{\expandafter\pld@firsttrue
                                \expandafter\pld@PLD
                                \expandafter{\pld@tempoly{#5}}}%
        \let\pld@va\pld@tempoly
        \pld@AddTo\pld@va{{-#5}}%
        \pld@MultiplyPoly\pld@current\pld@current\pld@va
    \fi
%    \end{macrocode}
% Then we initialize the variable's exponent and the two divisors and really
% start the algorithm.
%    \begin{macrocode}
    \pld@AddTo\pld@tempoly{1}%
    \@tempcnta\z@ \@tempcntb\@ne
    \pld@FindZeros{#1}{#3}%
%    \end{macrocode}
% Eventually we scan the remaining polynomial without multiplying the
% coefficient by the least common multiple of the denominators.
%    \begin{macrocode}
    \pld@false \pld@FactorizeInit
    \pld@FactorizeFinal}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@FactorizeFinal}
% Thus we might find nonrational zeros here. Let $a={}$|#1#2|, $b={}$|#6#7|,
% and $c={}$|#3#4|. Then we have to calculate
%   $\frac b{2a}\pm\sqrt{\frac{b^2}{4a^2}-\frac ca}$.
%    \begin{macrocode}
\def\pld@FactorizeFinal#1#2#3#4#5#6#7{%
    \ifnum\@multicnt=\tw@
        \pld@AddTo\pld@tempoly{1}%
        \pld@AccuSetX{#6}{#7}%
        \pld@AccuIfZero{\let\pld@va\@empty}%
                       {\pld@AccuMul12%
                        \pld@AccuMul{#2}{#1}%
                        \pld@AccuGet\pld@sub
                        \edef\pld@va{\noexpand\pld@R\pld@sub+}%
%    \end{macrocode}
% That's $\frac b{2a}$ so far, stored away in |\pld@va|.
%    \begin{macrocode}
                        \expandafter\pld@AccuMul\pld@sub}%
        \begingroup
          \pld@AccuSetX{#3}{#4}%
          \pld@AccuMul{-#2}{#1}%
          \pld@AccuGet\pld@temp
          \global\let\@gtempa\pld@temp
        \endgroup
        \expandafter\pld@AccuAdd\@gtempa
%    \end{macrocode}
% And now the accumulator holds $\frac{b^2}{4a^2}-\frac ca$. Depending on the
% sign---complex zeros are not supported, even though the complex analysis was
% my field of activity for some years---we do nothing more or get a printable
% version of the square root and \ldots
%    \begin{macrocode}
        \pld@AccuIfNegative
        {\@multicnt\tw@}%
        {\pld@AccuGet\pld@temp
         \expandafter\pld@FDefSqrt\pld@temp
%    \end{macrocode}
% append two nonrational zeros.
%    \begin{macrocode}
         \let\pld@vb\pld@va
         \pld@AddTo\pld@vb{\pld@R{-1}1}%
         \pld@Extend\pld@va{\pld@temp}%
         \pld@Extend\pld@vb{\pld@temp}%
         \expandafter\pld@AddZero\expandafter{\pld@va}%
         \expandafter\pld@AddZero\expandafter{\pld@vb}%
         \@multicnt\z@
        }%
    \fi
%    \end{macrocode}
% In this latter case or if the polynomial's degree has been zero from the
% beginning of this macro, we check whether we can omit the leading coeffient.
%    \begin{macrocode}
    \ifnum\@multicnt=\z@
         \pld@AccuSetX{#1}{#2}%
         \pld@AccuIfOne{\let\pld@current\@empty}%
                       {\def\pld@current{\pld@R{#1}{#2}}}%
    \fi
    \ifx\pld@current\@empty\else
        \let\pld@temp\pld@allines
        \def\pld@allines{\pld@PrintPolyWithDelims\pld@current}%
        \pld@Extend\pld@allines{\pld@temp}%
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@FDefSqrt}
% Finally we need a printable version of the square root of |#1|$/$|#2|.
% We use the fact, that the root is not rational, thus only one of nominator
% and denominator can be a square. Depending on the actual numbers, the
% submacro defines |\pld@temp| correctly. Note that this submacro is used to
% make the code more readable.
%    \begin{macrocode}
\def\pld@FDefSqrt#1#2{%
    \pld@IfSquare{#1}%
    {\pld@FDefSqrt@{\pld@R{\pld@temp}1}%
                   {\sqrt{\noexpand\pld@R{#2}1}}}%
    {\pld@IfSquare{#2}%
     {\ifnum\pld@temp=\@ne
          \pld@FDefSqrt@{\sqrt{\noexpand\pld@R{#1}1}}{}%
      \else
          \pld@FDefSqrt@{\sqrt{\noexpand\pld@R{#1}1}}%
                        {\pld@R{\pld@temp}1}%
      \fi}%
     {\def\pld@temp{\pld@F{\sqrt{\pld@R{#1}{#2}}}{}}}%
    }}
%    \end{macrocode}
% It just (e)defines a general fraction without expanding some control
% sequences.
%    \begin{macrocode}
\def\pld@FDefSqrt@#1#2{%
    \edef\pld@temp{\noexpand\pld@F
                   {\noexpand#1}%
                   {\ifx\@empty#2\@empty\else \noexpand#2\fi}}}
%    \end{macrocode}
% \end{macro}
%
%
% \section{Arithmetic}
%
% \begin{macro}{\pld@IfSquare}
% Let's begin with the macro used in the last section. As always, we initialize
% data.
%    \begin{macrocode}
\def\pld@IfSquare#1{%
    \@tempcnta=#1\relax
    \@multicnt\@tempcnta \@tempcntb\@tempcnta
    \divide\@tempcntb\tw@ \advance\@tempcntb\@ne
%    \end{macrocode}
% Then we use the iteration
%   $x_{n+1}=\left\lfloor \frac12\left(a+\lfloor\frac a{x_n}\rfloor\right)
%            \right\rfloor$
% to calculate $\lfloor \sqrt{|#1|}\rfloor$. In version 0.11 there was a bug
% in the loop condition.
%    \begin{macrocode}
    \loop \ifnum\@tempcntb<\@multicnt
        \@multicnt\@tempcntb
        \@tempcntb\@tempcnta
        \divide\@tempcntb\@multicnt
        \advance\@tempcntb\@multicnt
        \divide\@tempcntb\tw@
    \repeat
%    \end{macrocode}
% Now it is easy to decide whether |#1| is a square.
%    \begin{macrocode}
    \edef\pld@temp{\the\@multicnt}%
    \multiply\@multicnt\@multicnt
    \ifnum \@multicnt=\@tempcnta
        \expandafter\@firstoftwo
    \else
        \expandafter\@secondoftwo
    \fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@Euclidean}
% \begin{macro}{\pld@XEuclidean}
% \meta{macro}\marg{integer $a$}\marg{integer $b$}
% \begin{describe}
% The base of our rational arithmetic is the Euclidean algorithm. The contents
% of \meta{macro} becomes |{|$\frac a{\gcd(a,b)}$|}{|$\frac b{\gcd(a,b)}$|}|
% in the first case. The eXtended version adds the greatest common divisor:
% |{|$\frac a{\gcd(a,b)}$|}{|$\frac b{\gcd(a,b)}$|}{|$\gcd(a,b)$|}|.
% \end{describe}
% As described, the second version extends the first definition.
%    \begin{macrocode}
\def\pld@XEuclidean#1#2#3{\pld@Euclidean#1{#2}{#3}%
                          \edef#1{#1{\the\@tempcntb}}}
%    \end{macrocode}
% Here we assign the number smaller in size (in fact not bigger) to
% |\@tempcnta| and the other to |\@tempcntb|, and make both nonnegative.
%    \begin{macrocode}
\def\pld@Euclidean#1#2#3{%
   \@tempcnta#2\relax \divide\@tempcnta#3\relax
   \ifnum\@tempcnta=\z@ \@tempcnta#2\relax \@tempcntb#3\relax
                  \else \@tempcnta#3\relax \@tempcntb#2\relax \fi
   \ifnum\@tempcnta<\z@ \@tempcnta -\@tempcnta \fi
   \ifnum\@tempcntb<\z@ \@tempcntb -\@tempcntb \fi
%    \end{macrocode}
% The loop leaves the greatest common divisor in |\@tempcntb|.
%    \begin{macrocode}
   \pld@Euclidean@l
%    \end{macrocode}
% Now we only have to divide the numbers and define the macro |#1|.
%    \begin{macrocode}
   \@tempcnta#3\relax \divide\@tempcnta\@tempcntb
   \edef#1{{\the\@tempcnta}}%
   \@tempcnta#2\relax \divide\@tempcnta\@tempcntb
   \edef#1{{\the\@tempcnta}#1}}
%    \end{macrocode}
% And here is the usual Euclidean algorithm.\footnote{Note that
% \texttt{\bslash @multicnt} is used as a third scratch counter.}
%    \begin{macrocode}
\def\pld@Euclidean@l{%
   \ifnum\@tempcnta=\z@\else
       \@multicnt\@tempcntb
       \divide\@tempcntb\@tempcnta
       \multiply\@tempcntb\@tempcnta
       \advance\@multicnt -\@tempcntb
       \@tempcntb\@tempcnta
       \@tempcnta\@multicnt
       \expandafter\pld@Euclidean@l
   \fi}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@AccuGet}
% A rational number is stored as \marg{nominator}\marg{denominator} in
% the accumulator |\pld@accu|. Here we ensure that the denominator is
% positive.
%    \begin{macrocode}
\def\pld@AccuGet{\expandafter\pld@AccuGet@\pld@accu}
\def\pld@AccuGet@#1#2#3{%
    \ifnum #2<\z@ \edef#3{{-#1}{-#2}}\else\edef#3{{#1}{#2}}\fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AccuSet}
% \begin{macro}{\pld@AccuSetX}
% Setting the accumulator is also simple. We divide the nominator and
% denominator by their greatest common divisor only in the first case.
%    \begin{macrocode}
\def\pld@AccuSet#1#2{%
    \def\pld@accu{{#1}{#2}}%
    \expandafter\pld@Euclidean\expandafter\pld@accu\pld@accu
    \expandafter\pld@AccuGet@\pld@accu\pld@accu}
\def\pld@AccuSetX#1#2{\pld@AccuGet@{#1}{#2}\pld@accu}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@AccuPrint}
% Here we typeset the rational via |\frac| only if necessary.
%    \begin{macrocode}
\def\pld@AccuPrint{\expandafter\pld@AccuPrint@\pld@accu}
\def\pld@AccuPrint@#1#2{%
    \ifnum #2=\@ne \number#1\else \frac{\number#1}{\number#2}\fi}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AccuNegate}
% We just negate the nominator.
%    \begin{macrocode}
\def\pld@AccuNegate{\expandafter\pld@AccuNegate@\pld@accu}
\def\pld@AccuNegate@#1#2{\def\pld@accu{{-#1}{#2}}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AccuIfZero}
% \begin{macro}{\pld@AccuIfOne}
% \begin{macro}{\pld@AccuIfAbsOne}
% \begin{macro}{\pld@AccuIfNegative}
% All these definitions work the same way: expand |\pld@accu|, do the test,
% and either execute the first \meta{then} or the second \meta{else} part.
%    \begin{macrocode}
\def\pld@AccuIfZero{\expandafter\pld@AccuIfZero@\pld@accu}
\def\pld@AccuIfZero@#1#2{%
    \ifnum #1=\z@ \expandafter\@firstoftwo
            \else \expandafter\@secondoftwo \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@AccuIfOne{\expandafter\pld@AccuIfOne@\pld@accu}
\def\pld@AccuIfOne@#1#2{%
    \ifnum #1=#2\relax \expandafter\@firstoftwo
                 \else \expandafter\@secondoftwo \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@AccuIfAbsOne{\expandafter\pld@AccuIfAbsOne@\pld@accu}
\def\pld@AccuIfAbsOne@#1#2{%
    \ifnum #1=#2\relax \expandafter\@firstoftwo \else
        \ifnum -#1=#2\relax
            \expandafter\expandafter\expandafter\@firstoftwo
        \else
            \expandafter\expandafter\expandafter\@secondoftwo
        \fi
    \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@AccuIfNegative{\expandafter\pld@AccuIfNegative@\pld@accu}
\def\pld@AccuIfNegative@#1#2{%
    \ifnum #1<\z@ \@tempcnta\m@ne \else \@tempcnta\@ne \fi
    \ifnum #2<\z@ \@tempcnta -\@tempcnta \fi
    \ifnum \@tempcnta<\z@ \expandafter\@firstoftwo
                    \else \expandafter\@secondoftwo \fi}
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\pld@LCM}
% \marg{integer}
% \begin{describe}
% puts the least common multiple of \meta{integer} and \meta{nominator} into
% the accumulator.
% \end{describe}
% We use $\mathop{\mathrm{lcm}}(a,b)=\frac{a\cdot b}{\gcd(a,b)}=
% \frac{|\#1|}{\gcd(|\#1|,|\#3|)}\cdot |#3|$.
%    \begin{macrocode}
\def\pld@LCM{\expandafter\pld@LCM@\pld@accu}
\def\pld@LCM@#1#2#3{%
    \pld@Euclidean\pld@accu{#1}{#3}%
    \@tempcnta\expandafter\@firstoftwo\pld@accu\relax
    \multiply\@tempcnta#3\relax
    \edef\pld@accu{{\the\@tempcnta}1}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AccuMul}
% We use the Euclidean algorithm before \ldots
%    \begin{macrocode}
\def\pld@AccuMul{\expandafter\pld@AccuMul@\pld@accu}
\def\pld@AccuMul@#1#2#3#4{%
    \begingroup
      \pld@Euclidean\pld@va{#1}{#4}%
      \pld@Euclidean\pld@vb{#3}{#2}%
      \pld@ExpandTwo\pld@AccuMul@m\pld@va\pld@vb
      \xdef\@gtempa{{\the\@tempcnta}{\the\@tempcntb}}%
    \endgroup
    \let\pld@accu\@gtempa}
%    \end{macrocode}
% we multiply nominators and denominators.
%    \begin{macrocode}
\def\pld@AccuMul@m#1#2#3#4{%
    \@tempcnta#1\relax \multiply\@tempcnta#3\relax
    \@tempcntb#2\relax \multiply\@tempcntb#4\relax}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\pld@AccuAdd}
% The addition of two rationals is the most interesting part in this section.
% It is based upon the fact that $\frac ab+\frac cd=\frac{ad+bc}{bd}$ has
% \begin{eqnarray*}
% \meta{nominator}&=&\left(\textstyle\frac a{\gcd(a,c)}\cdot\frac d{\gcd(b,d)}+\frac b{\gcd(b,d)}\cdot\frac c{\gcd(a,c)}\right)\cdot\gcd(a,c),\\
% \meta{denominator}&=&\frac{bd}{\gcd(b,d)},
% \end{eqnarray*}
% where the factors and sums are all integers and potentially smaller in size
% than in $\frac{ad+bc}{db}$. As one quickly verifies\footnote{Sorry for that
% phrase, I'm a mathematician $:\!-)$}, the nominator and denominator has the
% greatest common divisor
%    \[\gcd(-\cdot-,b)\cdot\gcd\left(\textstyle-\cdot-,\frac d{\gcd(b,d)}\right),\]
% where $-\cdot-$ stands for the big parenthesized sum of the nominator.
%^^A The greatest common divisor is even \[\gcd(-\cdot-,b)\gcd(-\cdot-,d)\),
%^^A but we don't need either of this explicitly, the Euclidean algorithm will
%^^A take care of this.
%
% The implementation again expands |\pld@accu|, \ldots
%    \begin{macrocode}
\def\pld@AccuAdd{\expandafter\pld@AccuAdd@a\pld@accu}
%    \end{macrocode}
% and provides another submacro with the necessary fractions.
%    \begin{macrocode}
\def\pld@AccuAdd@a#1#2#3#4{%
    \ifnum#3=\z@\else
        \pld@AccuAdd@c{#1}{#2}{#3}{#4}%
    \fi}
\def\pld@AccuAdd@c#1#2#3#4{%
    \begingroup
      \pld@XEuclidean\pld@va{#1}{#3}%
      \pld@XEuclidean\pld@vb{#2}{#4}%
      \edef\pld@va{\pld@va\pld@vb}%
      \expandafter\pld@AccuAdd@b\pld@va{#2}{#4}}
%    \end{macrocode}
% We now have
% \begin{eqnarray*}
% \meta{nominator}&=&\left(|#1|\cdot|#5|+|#4|\cdot|#2|\right)\cdot |#3|,\\
% \meta{denominator}&=&|#7|\cdot|#5|.
% \end{eqnarray*}
%    \begin{macrocode}
\def\pld@AccuAdd@b#1#2#3#4#5#6#7#8{%
    \endgroup
    \@tempcnta#1\relax \multiply\@tempcnta#5\relax
    \@tempcntb#2\relax \multiply\@tempcntb#4\relax
    \advance\@tempcnta\@tempcntb
%    \end{macrocode}
% Finally we divide by $\gcd(-\cdot-,b)$ and multiply with
% $\frac{|\#3|}{|\#5|}$, which implicitly divides the result by
% $\gcd\left(-\cdot-,\frac d{\gcd(b,d)}\right)$.
%    \begin{macrocode}
    \expandafter\pld@Euclidean\expandafter\pld@accu\expandafter
        {\the\@tempcnta}{#7}%
    \pld@AccuMul{#3}{#5}}
%    \end{macrocode}
% \end{macro}
%
%
% \section{Horner's scheme}
%
% The following code lines come without comments. Good luck!
%
%    \begin{macrocode}
\def\pld@KVCases#1#2#3{%
    \@ifundefined{pld@#1@#2}%
    {\PackageError{Polynom}{Unknown value #2}{Try #3.}}%
    {\csname pld@#1@#2\endcsname}}
\def\pld@KVIf#1#2{%
    \@ifundefined{if#2}%
    {\PackageError{Polynom}{Unknown value #2}{Try `true' or `false'.}}%
    {\expandafter\let\expandafter#1\csname if#2\endcsname}}
%    \end{macrocode}
%    \begin{macrocode}
\define@key{pld}{showbase}[middle]{\pld@KVCases{showbase}{#1}{`false', `top', `middle', or `bottom'}}%
\def\pld@showbase@false{\let\pld@basepos=f}
\def\pld@showbase@top{\let\pld@basepos=t}
\def\pld@showbase@middle{\let\pld@basepos=m}
\def\pld@showbase@bottom{\let\pld@basepos=b}
\define@key{pld}{showvar}[true]{\pld@KVIf\pld@ifshowvar{#1}}
\define@key{pld}{showbasesep}[true]{\pld@KVIf\pld@ifshowbasesep{#1}}
\define@key{pld}{showmiddlerow}[true]{\pld@KVIf\pld@ifshowmiddlerow{#1}}

\define@key{pld}{resultstyle}{\def\pld@resultstyle{#1}}
\define@key{pld}{resultleftrule}[true]{\pld@KVIf\pld@ifhornerresultleftrule{#1}}
\define@key{pld}{resultrightrule}[true]{\pld@KVIf\pld@ifhornerresultrightrule{#1}}
\define@key{pld}{resultbottomrule}[true]{\pld@KVIf\pld@ifhornerresultbottomrule{#1}}

\define@key{pld}{tutor}[true]{\pld@KVCases{tutor}{#1}{`true', or `false'}}%
\def\pld@tutor@true{\let\pld@iftutor\iftrue}
\def\pld@tutor@false{\let\pld@iftutor\iffalse}
\define@key{pld}{tutorstyle}{\def\pld@tutorstyle{#1}}
\define@key{pld}{tutorlimit}{\@tempcnta#1\relax \advance\@tempcnta\@ne
    \edef\pld@tutorlimit{\the\@tempcnta}}

\define@key{pld}{equalcolwidths}[true]{\pld@KVIf\pld@ifhornerequalcolwidths{#1}}
\define@key{pld}{arraycolsep}{\def\pld@hornerarraycolsep{#1\relax}}
\define@key{pld}{arrayrowsep}{\def\pld@hornerarrayrowsep{#1\relax}}

\polyset{showbase,
         showvar=false,
         showbasesep=true,
         showmiddlerow=true,
         tutor=false,
         tutorlimit=1,
         tutorstyle=\scriptscriptstyle,
         resultstyle=,
         resultleftrule=false,
         resultrightrule=false,
         resultbottomrule=false,
         equalcolwidths=true,
         arraycolsep=\arraycolsep,
         arrayrowsep=.5\arraycolsep}

\define@key{pld}{mul}{\def\pld@mul{#1}}%
\define@key{pld}{plusface}{\pld@KVCases{hornerplusface}{#1}{`left' or 'right'}}%
\define@key{pld}{plusyoffset}{\@tempdima#1\relax \edef\pld@hornerplusyoffset{\the\@tempdima}}
\define@key{pld}{downarrowxoffset}{\@tempdima#1\relax \edef\pld@hornerdownarrowxoffset{\the\@tempdima}}
\define@key{pld}{diagarrowxoffset}{\@tempdima#1\relax \edef\pld@hornerdiagarrowxoffset{\the\@tempdima}}
\define@key{pld}{downarrow}{\def\pld@hornerdownarrow{#1}}
\define@key{pld}{diagarrow}{\def\pld@hornerdiagarrow{#1}}
\def\pld@hornerplusface@left{\let\pld@hornerplusface\llap}
\def\pld@hornerplusface@right{\let\pld@hornerplusface\rlap}
\polyset{mul=\cdot,
         plusface=right,
         plusyoffset=\z@,
         downarrowxoffset=\z@,
         diagarrowxoffset=\z@,
         downarrow={\vector(0,-1){2.5}},
         diagarrow={\vector(2,1){1.6}}}
%    \end{macrocode}
%    \begin{macrocode}
\newcommand*\polyhornerscheme[1][]{%
    \begingroup
    \let\pld@stage\maxdimen \polyset{#1}%
    \pld@GetPoly{\pld@polya}%
                {\expandafter\pld@Horner\expandafter{\pld@polya}%
    \endgroup \ignorespaces}}

\def\pld@Horner#1{%
    \pld@GetTotalDegree\pld@degree{#1}%
    \pld@Horner@#1++%
    \pld@ArrangeHorner}

\def\pld@Horner@#1+{%
    \pld@SplitMonom\pld@HornerInit{#1}%
    \pld@HornerIterate}

\def\pld@HornerIterate#1+{%
    \advance\@tempcnta\m@ne
    \ifx\@empty#1\@empty
        \ifnum \@tempcnta<\z@
            \let\pld@next\@empty
        \else
            \pld@HornerStep{\pld@R01}{}%
            \def\pld@next{\pld@HornerIterate+}%
        \fi
    \else
        \pld@GetTotalDegree\pld@degree{#1}%
        \ifnum \pld@degree=\@tempcnta
            \pld@SplitMonom\pld@HornerStep{#1}%
            \let\pld@next\pld@HornerIterate
        \else
            \pld@HornerStep{\pld@R01}{}%
            \def\pld@next{\pld@HornerIterate#1+}%
        \fi
    \fi
    \pld@next}

\def\pld@HornerStep#1#2{%
    \pld@AddTo\pld@lastline{&\pld@PrintPolyArg{#1}}%
    \pld@MultiplyPoly\pld@lastsum\pld@lastsum\pld@value
    \pld@Simplify\pld@lastsum
    \ifx\pld@lastsum\@empty \def\pld@lastsum{\pld@R 01}\fi
    \pld@AddTo\pld@subline{&}%
    \pld@HornerExtendLine\pld@subline
    \pld@AddTo\pld@lastsum{+#1}%
    \pld@Simplify\pld@lastsum
    \pld@AddTo\pld@currentline{&}%
    \pld@iftutor
        \pld@HornerExtendCurrentLine
        \advance\@multicnt\@ne
        \pld@HornerIfTutorStage{\pld@HornerExtendTutor\pld@HornerOtherDown}%
        \advance\@multicnt\m@ne
        \ifnum\@tempcnta>\z@
            \pld@HornerIfTutorStage{\pld@HornerExtendTutor\pld@HornerDiag}%
        \fi
    \else
        \pld@HornerExtendCurrentLine
    \fi}

\def\pld@HornerExtendTutor#1{%
    \ifnum\@tempcnta=\z@ \pld@AddTo\pld@hornerresult#1%
                   \else \pld@AddTo\pld@currentline#1\fi}
\def\pld@HornerExtendCurrentLine{%
    \ifnum\@tempcnta=\z@
        \let\pld@hornerresult\@empty
        \pld@Extend\pld@hornerresult{\expandafter{\expandafter\pld@resultstyle\expandafter{%
                                     \expandafter\pld@PrintPolyArg\expandafter{\pld@lastsum}}}}%
        \pld@HornerIfStage{}%
                          {\let\pld@lastsum\@empty
                           \pld@Extend\pld@lastsum{\expandafter\phantom\expandafter{\pld@hornerresult}}%
                           \let\pld@hornerresult\pld@lastsum}%
        \expandafter\@gobbletwo
    \fi
    \pld@HornerExtendLine\pld@currentline}
\def\pld@HornerExtendLine#1{%
    \pld@HornerIfStage{\pld@Extend#1{\expandafter\pld@PrintPolyArg\expandafter{\pld@lastsum}}}%
                      {\pld@Extend#1{\expandafter\phantom\expandafter{%
                                     \expandafter\pld@PrintPolyArg\expandafter{\pld@lastsum}}}}%
}

\def\pld@HornerFirstDown{%
    \rlap{\kern\pld@hornerdownarrowxoffset\relax
          \unitlength\ht\@arstrutbox
          \begin{picture}(0,0)%
          \setbox\z@\hbox{$\pld@tutorstyle{\pld@hornerdownarrow}$}%
          \put(0,.5){\raise\ht\z@\hbox{\raise\dp\z@\copy\z@}}%
          \end{picture}}}
\def\pld@HornerOtherDown{%
    \pld@HornerFirstDown
    \pld@hornerplusface{\kern\pld@hornerdownarrowxoffset\relax
                        \smash{\raise\pld@hornerplusyoffset
                               \hbox{\raise.5\ht\@arstrutbox
                                     \vbox to 2.5\ht\@arstrutbox
                                     {\vss$\pld@tutorstyle{+}$\vss}}}}}
\def\pld@HornerDiag{%
    \rlap{\kern\pld@hornerdiagarrowxoffset\relax
        \unitlength\ht\@arstrutbox
        \begin{picture}(0,0)%
        \setbox\z@\hbox{$\pld@tutorstyle{\pld@hornerdiagarrow}$}%
        \put(0,.5){\box\z@}%
        \put(0,.5){\kern.55\ht\@arstrutbox
                   $\pld@tutorstyle{\pld@mul \pld@hornerleftdelim
                                    \pld@PrintPolyWithDelims\pld@value
                                    \pld@hornerrightdelim}$}%
        \end{picture}}}

\def\pld@HornerIfStage{%
    \advance\@multicnt\m@ne
    \ifnum\@multicnt>\z@ \expandafter\@firstoftwo
                   \else \expandafter\@secondoftwo \fi}
\def\pld@HornerIfTutorStage{%
    \ifnum\@multicnt>\@ne
        \ifnum\@multicnt>\pld@tutorlimit
            \expandafter\expandafter\expandafter\@gobble
        \else
            \expandafter\expandafter\expandafter\@firstofone
        \fi
    \else
        \expandafter\@gobble
    \fi}

\def\pld@HornerInit#1#2{%
    \let\pld@V\@firstoftwo
    \ifx\@empty#2\@empty\else
        \edef\pld@var{#2}%
        \@ifundefined{pld@value@\pld@var}%
        {\PackageError{Polynom}{Missing value for variable \pld@var}{}%
         \@namedef{pld@value@\pld@var}{\pld@R01}}%
        {}%
        \expandafter\let\expandafter\pld@value\csname pld@value@\pld@var\endcsname
    \fi
%
    \setbox\@tempboxa\hbox{$\pld@PrintPoly\pld@value$}%
    \pld@ifminus
        \let\pld@hornerleftdelim(%
        \let\pld@hornerrightdelim)%
    \else
        \let\pld@hornerleftdelim\@empty
        \let\pld@hornerrightdelim\@empty
    \fi
%
    \@multicnt\pld@stage\relax
    \@tempcnta\pld@degree\relax
    \def\pld@lastline{\pld@PrintPolyArg{#1}}%
    \let\pld@subline\@empty
    \let\pld@currentline\@empty
    \def\pld@lastsum{#1}%
    \pld@iftutor
        \pld@HornerExtendCurrentLine
        \advance\@multicnt\@ne
        \pld@HornerIfTutorStage{\pld@AddTo\pld@currentline\pld@HornerFirstDown}%
        \advance\@multicnt\m@ne
        \ifnum\@tempcnta>\z@
            \pld@HornerIfTutorStage{\pld@AddTo\pld@currentline\pld@HornerDiag}%
        \fi
    \else
        \pld@HornerExtendCurrentLine
    \fi
    \def\pld@lastsum{#1}%
    \@tempcnta\pld@degree\relax % init moved up, delete this?
    \advance\@tempcnta\thr@@
    \edef\pld@hornermaxcol{\the\@tempcnta}%
    \@tempcnta\pld@degree\relax}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@ArrangeHorner{%
    \begingroup
    \@tempdima\z@
    \pld@MeasureCells\pld@lastline
    \pld@MeasureCells\pld@subline
    \pld@MeasureCells\pld@currentline
    \pld@MeasureCells\pld@hornerresult
    \everycr{}\tabskip\z@skip
    \@tempdimb\ht\strutbox \advance\@tempdimb\pld@hornerarrayrowsep
    \@tempdimc\dp\strutbox \advance\@tempdimc\pld@hornerarrayrowsep
    \setbox\@arstrutbox\hbox{\vrule \@height\@tempdimb
                                    \@depth\@tempdimc
                                    \@width\z@}%
    \pld@ifhornerequalcolwidths\else
        \def\@startpbox##1{\hfil\vtop\bgroup \hbox\bgroup \@arrayparboxrestore}%
        \def\@endpbox{\@finalstrut\@arstrutbox \egroup\par\egroup}%
    \fi
    \def\pld@leftdelim{(}\def\pld@rightdelim{)}%
    \leavevmode
    \hbox{$\vcenter{\offinterlineskip \arraycolsep\pld@hornerarraycolsep
        \halign{\@arstrut
                \hskip\arraycolsep \hfill\ensuremath{##}\hskip\arraycolsep
              &##&&%
                \hskip\arraycolsep \@startpbox\@tempdima\hfill\ensuremath{##}\@endpbox \hskip\arraycolsep\cr
                \pld@ShowBase t&\pld@ifshowbasesep\vrule\fi&\pld@lastline\cr
                \pld@ifshowmiddlerow \pld@ShowBase m&\pld@ifshowbasesep\vrule\fi&\pld@subline\cr \fi \cline{2-\pld@hornermaxcol}%
                \pld@ShowBase b&&\pld@currentline\omit
                \pld@ifhornerresultleftrule \vrule \fi
                \hskip\arraycolsep \@startpbox\@tempdima\relax\hfill\ensuremath{\pld@hornerresult}\@endpbox \hskip\arraycolsep
                \pld@ifhornerresultrightrule \vrule \fi \cr
                \pld@ifhornerresultbottomrule \cline{\pld@hornermaxcol-\pld@hornermaxcol} \fi
               }%
    }$}%
    \endgroup}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@ShowBase#1{%
    \ifx#1\pld@basepos
        \pld@ifshowvar x=\fi\pld@PrintPoly\pld@value
    \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@MeasureCells#1{\expandafter\pld@MeasureCells@#1&\@nil&}
\def\pld@MeasureCells@#1&{%
    \ifx\@nil#1\relax\else
        \setbox\@tempboxa\hbox{\ensuremath{#1}}%
        \ifdim\wd\@tempboxa>\@tempdima
            \@tempdima\wd\@tempboxa
        \fi
        \expandafter\pld@MeasureCells@
    \fi}
%    \end{macrocode}
%    \begin{macrocode}
\def\pld@GetTotalDegree#1#2{%
    \begingroup
    \let\pld@R\@gobbletwo \let\pld@F\@gobbletwo \let\pld@S\@gobbletwo
    \def\pld@V##1##2{\advance\@tempcnta##2\relax}%
    \def#1##1+##2\@nil{##1}%
    \edef#1{\@tempcnta\z@#1#2+\@nil}%
    #1\xdef\@gtempa{\the\@tempcnta}%
    \endgroup
    \let#1\@gtempa}
%    \end{macrocode}
%
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
%
% \begingroup\small
% \section{History}
% \renewcommand\labelitemi{--}
% \begin{itemize}
% \item[0.1] from 2000/04/18 (private test version)
%   \item long division algorithm et al, basic scanner, basic simplification
% \item[0.11] from 2001/03/23
%   \item total reimplementation except division algorithm et al
%   \item improved: scanner, simplification, handling of symbols
%   \item new: gcd, factorization, rational arithmetic, key $=$ value interface
% \item[0.12] from 2001/04/11
%   \item bugs in |\pld@IfSquare| and |\pld@ScanOpen| removed
%   \item slightly improved scanner (|^| on expressions) and new key \texttt{delims}
% \item[0.13] from 2001/09/27
%   \item new \texttt{stage} key allows stepwise printing of polynomial long divisions
% \item[0.14] from 2002/01/10
%   \item added \texttt{style=C}; this led to the new \texttt{div} key and the optional argument of \texttt{delims}
% \item[0.15] from 2002/10/29
%   \item bugs fixed in |\polygcd| and |\pld@LongEuclideanPoly|
% \item[0.16] from 2004/08/12
%   \item added (bugfixed version of) Horner's scheme and fixed bug in |\pld@InsertItems@find|
% \end{itemize}
% The phrase `et al' stands for the definitions directly related to the
% division algorithm: polynomial multiplication, |\pld@IfNeedsDivision|,
% subtraction, and alignment.
% \medskip
%
% \noindent TODO:
% \begin{itemize}
%   \item PBZ
%   \item use \texttt{stage} also on \cs{polylonggcd}
%   \item possibility to highlight the most recent \texttt{stage}
%   \item remove problems inside array and tabular
%   \item carry out dependencies in the implementation part (or remove them)
%   \item internal data format: introduce linear, square factors?
%   \item generalize exponents for printing $y^{(4)}-y^{(2)}+\ldots$ ?
%   \item define derivatives?
% \end{itemize}
% \endgroup
%
%
% \Finale
%
%%
%%
\endinput