path: root/Master/texmf-dist/tex/latex/polexpr/polexpr.sty

% author: Jean-François Burnol
% License: LPPL 1.3c (author-maintained)
\ProvidesPackage{polexpr}%
  [2019/02/12 v0.7.4 Polynomial expressions with rational coefficients (JFB)]%
\RequirePackage{xintexpr}[2018/06/17]% xint 1.3c for \ifxintglobaldefs boolean
\edef\POL@restorecatcodes
    {\catcode`\noexpand\_ \the\catcode`\_ %
     \catcode`\noexpand\! \the\catcode`\! %
     \catcode`\noexpand\* \the\catcode`\* %
     \catcode0 \the\catcode0\relax}%
\catcode`\_ 11 \catcode0 12 \catcode`\* 12
\long\def\xint_stop_atfirstoftwo  #1#2{ #1}% not yet in xint 1.3c
\long\def\xint_stop_atsecondoftwo #1#2{ #2}%

%% PATCH xintexpr TO AUTHORIZE ' IN NAMES (0.5.1)
\catcode`\! 11
\def\POL@XINT_expr_scanfunc_b #1%
{%
  \ifx !#1\xint_dothis{(_}\fi
  \ifcat \relax#1\xint_dothis{(_}\fi
  \if (#1\xint_dothis{\xint_firstoftwo{(`}}\fi
  \if '#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \if @#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \if _#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \ifnum \xint_c_ix<1\string#1 \xint_dothis \XINT_expr_scanfunc_a \fi
  \ifcat a#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \xint_orthat {(_}%
    #1%
}%

%% AUXILIARIES
\catcode`! 3
%% added at 0.7
\newcommand\polexprsetup[1]{\POL@setup_parsekeys #1,=!,\xint_bye}%
\def\POL@setup_parsekeys #1=#2#3,{%
    \ifx!#2\expandafter\xint_bye\fi
    \csname POL@setup_setkey_\xint_zapspaces #1 \xint_gobble_i\endcsname
    \xint_firstoftwo
    {\PackageWarning{polexpr}{The \detokenize{#1} key is unknown! ignoring}}%
    {\xintZapLastSpaces{#2#3}}%
    \POL@setup_parsekeys
}%
\catcode`! 11 
\def\POL@setup_setkey_norr     #1#2{\edef\POL@norr}%
\def\POL@setup_setkey_sqfnorr  #1#2{\edef\POL@sqfnorr}%
\polexprsetup{norr=_norr, sqfnorr=_sqf_norr}

\newcount\POL@count
\newif\ifPOL@pol
\newif\ifxintveryverbose
\newif\ifpoltypesetall
\newif\ifPOL@tosturm@makefirstprimitive
\POL@tosturm@makefirstprimitivetrue
\newif\ifPOL@isolz@nextwillneedrefine
\newif\ifpoltoexprall
%% the main exchange structure (stored in macros \POLuserpol@<name>)
%% is: degree.\empty{coeff0}{coeff1}....{coeffN}
%% (degree=N except zero polynomial recognized from degree set to -1
%%  but it has always the {0/1[0]} coeff0.)
\def\POL@ifZero#1{\expandafter\POL@ifZero@aux#1;}%
\def\POL@ifZero@aux #1#2;{\if-#1\expandafter\xint_firstoftwo
                             \else\expandafter\xint_secondoftwo
                          \fi}%
\def\POL@split#1.#2;#3#4% separates degree and list of coefficients
% The \expandafter chain removes the \empty token
   {\def#3{#1}\expandafter\def\expandafter#4\expandafter{#2}}%
%
\def\POL@resultfromarray #1{% ATTENTION, **MUST** be executed with
% \count@ set to 1 + degree (\count@ = 0 for zero polynomial)
    \edef\POL@result{\ifnum\count@>\z@
      \the\numexpr\count@-\@ne.\noexpand\empty
      \xintiloop [1+1]%
      \expandafter\POL@braceit\csname POL@array#1\xintiloopindex\endcsname
      \ifnum\xintiloopindex<\count@
      \repeat
      \else-1.\noexpand\empty{0/1[0]}\fi}%
}%
\def\POL@braceit#1{{#1}}% needed as \xintiloopindex can not "see" through braces


\newcommand\PolDef[3][x]{\poldef #2(#1):=#3;}%
\def\poldef{\edef\POL@restoresemicolon{\catcode59=\the\catcode59\relax}%
                \catcode59 12 \POL@defpol}%
\def\POL@defpol #1(#2)#3=#4;{%
   \POL@restoresemicolon
   \edef\POL@tmp{\ifxintverbose1\else0\fi}%
   \unless\ifxintveryverbose\xintverbosefalse\fi
   \let\POL@originalXINT_expr_scanfunc_b\XINT_expr_scanfunc_b
   \let\XINT_expr_scanfunc_b\POL@XINT_expr_scanfunc_b
   \xintdeffunc __pol(#2):=0+(#4);% force conversion to raw if a constant
   \if1\POL@tmp\xintverbosetrue\fi
   \edef\POL@polname{\xint_zapspaces #1 \xint_gobble_i}%
   \let\XINT_expr_scanfunc_b\POL@originalXINT_expr_scanfunc_b
   \begingroup
     \setbox0\hbox{%
     \let\xintScalarAdd\xintAdd
     \let\xintScalarSub\xintSub
     \let\xintScalarMul\xintMul
     \let\xintScalarDiv\xintDiv
     \let\xintScalarPow\xintPow
     \let\xintScalarOpp\xintOpp
     \let\xintAdd\POL@add
     \let\xintMul\POL@mul
     \let\xintDiv\POL@div
     \let\xintPow\POL@pow
     \let\xintOpp\POL@opp
     \def\xintSub ##1##2{\xintAdd{##1}{\xintOpp{##2}}}%
     % \xintAdd{0} to get \POL@result defined even if numerical only expression
     % I could also test \ifPOL@pol, but this is anyhow small overhead
     \xintAdd{0}%
       {\csname XINT_expr_userfunc___pol\endcsname
        {\global\POL@poltrue\def\POL@result{1.\empty{0/1[0]}{1/1[0]}}}}%
     \expandafter}\expandafter
   \endgroup\expandafter
   \def\csname POLuserpol@\POL@polname\expandafter\endcsname
        \expandafter{\POL@result}%
   \expandafter\POL@newpol\expandafter{\POL@polname}%
}%
%%
\def\POL@newpol#1{%
   \expandafter\POL@ifZero\csname POLuserpol@#1\endcsname
      {\@namedef{XINT_expr_userfunc_#1}##1{0/1[0]}}%
      {\POL@newpolhorner{#1}}%
   \expandafter\XINT_expr_defuserfunc
     \csname XINT_expr_func_#1\endcsname{#1}{expr}%
   \expandafter\let\csname XINT_flexpr_func_#1\endcsname\@undefined
   \ifxintverbose\POL@info{#1}\fi
}%
\def\POL@newfloatpol#1{%
   \expandafter\POL@ifZero\csname POLuserpol@#1\endcsname
      {\@namedef{XINT_flexpr_userfunc_#1}##1{0[0]}}%
      {\POL@newfloatpolhorner{#1}}%
   \expandafter\XINT_expr_defuserfunc
     \csname XINT_flexpr_func_#1\endcsname{#1}{flexpr}%
   \ifxintverbose\POL@floatinfo{#1}\fi
}%
\def\POL@info #1{%
   \xintMessage {polexpr}{Info}%
        {Function #1 for the \string\xintexpr\space parser is
         associated to \string\XINT_expr_userfunc_#1\space
         whose meaning uses Horner scheme:
         \expandafter\meaning
         \csname XINT_expr_userfunc_#1\endcsname}%
}%
\def\POL@floatinfo #1{%
   \xintMessage {polexpr}{Info}%
        {Function #1 for the \string\xintfloatexpr\space parser is
         associated to \string\XINT_flexpr_userfunc_#1\space
         whose meaning uses Horner scheme:
         \expandafter\meaning
         \csname XINT_flexpr_userfunc_#1\endcsname}%
}%
%
\def\POL@newpolhorner#1{%
   %% redefine function to expand by Horner scheme. Is this useful?
   %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10?
% note: I added {0/1[0]} item to zero polynomial also to facilitate this
   \expandafter\expandafter\expandafter\POL@split
      \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
   \edef\POL@var@coeffs{\xintRevWithBraces{\POL@var@coeffs}}%
   \begingroup
      \expandafter\POL@newpol@horner\POL@var@coeffs\relax
   \expandafter
   \endgroup
   \expandafter\def\csname XINT_expr_userfunc_#1\expandafter\endcsname
      \expandafter##\expandafter1\expandafter{\POL@tmp{##1}}%
}%
\def\POL@newfloatpolhorner#1{%
   %% redefine function to expand by Horner scheme. Is this useful?
   %% perhaps bad idea for numerical evaluation of thing such as (1+x)^10?
% note: I added {0/1[0]} item to zero polynomial also to facilitate this
   \expandafter\expandafter\expandafter\POL@split
      \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
   \edef\POL@var@coeffs{\xintRevWithBraces{\POL@var@coeffs}}%
   \begingroup
      \expandafter\POL@newpol@floathorner\POL@var@coeffs\relax
   \expandafter
   \endgroup
   \expandafter\def\csname XINT_flexpr_userfunc_#1\expandafter\endcsname
      \expandafter##\expandafter1\expandafter{\POL@tmp{##1}}%
}%
\def\POL@newpol@horner#1{\let\xintAdd\relax\let\xintMul\relax
                  \def\POL@tmp##1{#1}\POL@newpol@horner@loop.}%
\def\POL@newpol@horner@loop.#1{%
  \if\relax#1\expandafter\xint_gob_til_dot\fi
  \edef\POL@tmp##1{\xintiiifZero{#1}
                    {\@firstofone}{\xintAdd{#1}}%
                    {\xintMul{##1}{\POL@tmp{##1}}}}%
  \POL@newpol@horner@loop.%
}%
\def\POL@newpol@floathorner#1{\let\XINTinFloatAdd\relax\let\XINTinFloatMul\relax
                  \def\xintAdd{\XINTinFloatAdd}\def\xintMul{\XINTinFloatMul}%
                  \edef\POL@tmp##1{\XINTinFloatdigits{#1}}%
                  \POL@newpol@floathorner@loop.}%
\def\POL@newpol@floathorner@loop.#1{%
  \if\relax#1\expandafter\xint_gob_til_dot\fi
  \edef\POL@tmp##1{\xintiiifZero{#1}
                    {\@firstofone}{\xintAdd{\XINTinFloatdigits{#1}}}%
                    {\xintMul{##1}{\POL@tmp{##1}}}}%
  \POL@newpol@floathorner@loop.%
}%


\newcommand\PolGenFloatVariant[1]{\POL@newfloatpol{#1}}%


\newcommand\PolLet[2]{\if=\noexpand#2\expandafter\xint_firstoftwo
                      \else\expandafter\xint_secondoftwo\fi
                      \POL@@let\POL@let{#1}{#2}}%
\def\POL@@let#1#2#3{\POL@let{#1}{#3}}%
\def\POL@let#1#2{%
    \expandafter\let\csname POLuserpol@#1\expandafter\endcsname
                    \csname POLuserpol@#2\endcsname
    \expandafter\let\csname XINT_expr_userfunc_#1\expandafter\endcsname
                    \csname XINT_expr_userfunc_#2\endcsname
    \expandafter\XINT_expr_defuserfunc
       \csname XINT_expr_func_#1\endcsname{#1}{expr}%
    \ifxintverbose\POL@info{#1}\fi
}%
\newcommand\PolGlobalLet[2]{\begingroup
    \globaldefs\@ne
    \if=\noexpand#2\expandafter\xint_firstoftwo
    \else\expandafter\xint_secondoftwo\fi
% do I need to check something here relative to \xintNewExpr?
    \POL@@globallet\POL@globallet {#1}{#2}}%
\def\POL@@globallet#1#2#3{\POL@globallet{#1}{#3}}%
\def\POL@globallet#1#2{\POL@let{#1}{#2}\endgroup}%

\newcommand\PolAssign[1]{\def\POL@polname{#1}\POL@assign}% zap spaces in #1?
\def\POL@assign#1\toarray#2{%
    \expandafter\expandafter\expandafter\POL@split
    \csname POLuserpol@\POL@polname\endcsname;\POL@var@deg\POL@var@coeffs
    \xintAssignArray\POL@var@coeffs\to#2%
    % modify \#200 macro to return 0/1[0] for out of range indices
    \@namedef{\xint_arrayname00}##1##2##3{%
    \@namedef{\xint_arrayname00}####1{%
        \ifnum####1>##1 \xint_dothis{ 0/1[0]}\fi
        \ifnum####1>\m@ne \xint_dothis
                {\expandafter\expandafter\expandafter##3%
                 \csname##2####1\endcsname}\fi
        \unless\ifnum-####1>##1 \xint_dothis
                {\expandafter\expandafter\expandafter##3%
                 \csname##2\the\numexpr##1+####1+\@ne\endcsname}\fi
        \xint_orthat{ 0/1[0]}}% space stops a \romannumeral0
    }%
    \csname\xint_arrayname00\expandafter\expandafter\expandafter\endcsname
      \expandafter\expandafter\expandafter
      {\csname\xint_arrayname0\expandafter\endcsname\expandafter}\expandafter
      {\xint_arrayname}{ }%
}%


\newcommand\PolGet{}%
\def\PolGet#1#2\fromarray#3{%
    \begingroup % closed in \POL@getfromarray
    \POL@getfromarray{#1}{#3}%
    \POL@newpol{#1}%
}%
\def\POL@getfromarray#1#2{%
        \count@=#2{0} %<- intentional space
  \ifnum\count@=\z@
     \def\POL@result{-1.\empty{0/1[0]}}% 0.5 fix for empty array
  \else
        \xintloop
          \edef\POL@tmp{#2{\count@}}%
          \edef\POL@tmp{\xintRaw{\POL@tmp}}%
% sadly xinttools (current 1.3a) arrays have no setters for individual items... 
          \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
        \if0\xintiiSgn{\POL@tmp}%
            \advance\count@\m@ne
        \repeat
% dans le cas particulier d'un array avec que des éléments nuls, \count@ est
% ici devenu 0 et la boucle s'est arrêtée car #2{0} était au moins 1. De plus
% \POL@tmparray1 est bien 0/1[0] donc ok pour polynôme nul dans \POL@result
        \count\tw@\count@
        \xintloop
% on mouline tous les coeffs via \xintRaw
        \ifnum\count@>\@ne
          \advance\count@\m@ne
          \edef\POL@tmp{#2{\count@}}%
          \edef\POL@tmp{\xintRaw{\POL@tmp}}%
          \expandafter\let\csname POL@tmparray\the\count@\endcsname\POL@tmp
        \repeat
        \count@\count\tw@          
        \def\POL@tmp##1.{{\csname POL@tmparray##1\endcsname}}%
        \edef\POL@result{\the\numexpr\count@-\@ne.\noexpand\empty
                         \xintiloop[1+1]%
                         \expandafter\POL@tmp\xintiloopindex.%
                         \ifnum\xintiloopindex<\count@
                         \repeat}%
  \fi
    \expandafter
    \endgroup
    \expandafter
    \def\csname POLuserpol@#1\expandafter\endcsname
        \expandafter{\POL@result}%
}%


\newcommand\PolFromCSV[2]{%
    \begingroup % closed in \POL@getfromarray
    \xintAssignArray\xintCSVtoList{#2}\to\POL@arrayA
    \POL@getfromarray{#1}\POL@arrayA
    \POL@newpol{#1}%
% semble un peu indirect et sous-optimal
% mais je veux élaguer les coefficients nuls. Peut-être à revoir.
}%


\newcommand\PolTypesetCmdPrefix[1]{\xintiiifSgn{#1}{}{+}{+}}%
\newcommand\PolTypesetCmd[1]{\xintifOne{\xintiiAbs{#1}}%
                               {\ifnum\PolIndex=\z@\xintiiSgn{#1}\else
                                \xintiiifSgn{#1}{-}{}{}\fi
                                \let\PolIfCoeffIsPlusOrMinusOne\@firstoftwo}%
                               {\PolTypesetOne{#1}%
                                \let\PolIfCoeffIsPlusOrMinusOne\@secondoftwo}%
                             }%
\newcommand\PolTypesetOne{\xintSignedFrac}%
\newcommand\PolTypesetMonomialCmd{%
  \ifcase\PolIndex\space
  %
  \or\PolVar
  \else\PolVar^{\PolIndex}%
  \fi
}%
\newcommand\PolTypeset{\@ifstar
  {\def\POL@ts@ascending{1}\POL@Typeset}%
  {\def\POL@ts@ascending{0}\POL@Typeset}%
}%
\newcommand\POL@Typeset[2][x]{% LaTeX \newcommand forces optional argument first
   \ensuremath{%
        \expandafter\expandafter\expandafter\POL@split
        \csname POLuserpol@#2\endcsname;\POL@var@deg\POL@var@coeffs
        \if\POL@ts@ascending1%
           \def\PolIndex{0}%
           \let\POL@ts@reverse\@firstofone
           \let\POL@@ne@or@m@ne\@ne
        \else
           \let\PolIndex\POL@var@deg
           \ifnum\PolIndex<\z@\def\PolIndex{0}\fi
           \let\POL@ts@reverse\xintRevWithBraces
           \let\POL@@ne@or@m@ne\m@ne
        \fi
        \def\PolVar{#1}%
        \ifnum\POL@var@deg<\z@
            \PolTypesetCmd{0/1[0]}\PolTypesetMonomialCmd
        \else
         \ifnum\POL@var@deg=\z@
           \expandafter\PolTypesetCmd\POL@var@coeffs\PolTypesetMonomialCmd
         \else
           \def\POL@ts@prefix##1{\let\POL@ts@prefix\PolTypesetCmdPrefix}%
           \expandafter\POL@ts@loop
              \romannumeral-`0\POL@ts@reverse{\POL@var@coeffs}\relax
         \fi
        \fi
   }%
}%
\def\POL@ts@loop{\ifpoltypesetall\expandafter\xint_firstoftwo
                 \else\expandafter\xint_secondoftwo\fi
    {\POL@ts@nocheck}{\POL@ts@check}.%
}%
\def\POL@ts@check.#1{%
  \if\relax#1\expandafter\xint_gob_til_dot\fi
  \xintiiifZero{#1}%
      {}%
      {\POL@ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd}%
  \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne@or@m@ne}\POL@ts@check.%
}%
\def\POL@ts@nocheck.#1{%
  \if\relax#1\expandafter\xint_gob_til_dot\fi
  \POL@ts@prefix{#1}\PolTypesetCmd{#1}\PolTypesetMonomialCmd
  \edef\PolIndex{\the\numexpr\PolIndex+\POL@@ne@or@m@ne}\POL@ts@nocheck.%
}%


\newcommand\PolMapCoeffs[2]{% #1 = macro, #2 = name
    \POL@mapcoeffs{#1}{#2}%
    \POL@newpol{#2}%
}%
\def\POL@mapcoeffs#1#2{%
    \begingroup
       \def\POL@mapcoeffs@macro{#1}%
       \expandafter\expandafter\expandafter\POL@split
         \csname POLuserpol@#2\endcsname;\POL@mapcoeffs@deg\POL@mapcoeffs@coeffs
% ATTENTION à ne pas faire un \expandafter ici, car brace removal si 1 item
       \xintAssignArray\POL@mapcoeffs@coeffs\to\POL@arrayA
       \def\index{0}%
       \count@\z@
       \expandafter\POL@map@loop\expandafter.\POL@mapcoeffs@coeffs\relax
       \xintloop
% this abuses that \POL@arrayA0 is never 0.
       \xintiiifZero{\csname POL@arrayA\the\count@\endcsname}%
         {\iftrue}%
         {\iffalse}%
       \advance\count@\m@ne
       \repeat
% donc en sortie \count@ est 0 ssi pol nul.
       \POL@resultfromarray A%
     \expandafter
    \endgroup
    \expandafter
    \def\csname POLuserpol@#2\expandafter\endcsname\expandafter{\POL@result}%
}%
\def\POL@map@loop.#1{\if\relax#1\expandafter\xint_gob_til_dot\fi
    \advance\count@\@ne
    \edef\POL@map@coeff{\POL@mapcoeffs@macro{#1}}%
    \expandafter
    \let\csname POL@arrayA\the\count@\endcsname\POL@map@coeff
    \edef\index{\the\numexpr\index+\@ne}%
    \POL@map@loop.}%
\def\POL@xintIrr#1{\xintIrr{#1}[0]}%
\newcommand\PolReduceCoeffs{\@ifstar\POL@sreducecoeffs\POL@reducecoeffs}%
\def\POL@reducecoeffs#1{\PolMapCoeffs{\POL@xintIrr}{#1}}%
\def\POL@sreducecoeffs#1{\PolMapCoeffs{\xintPIrr}{#1}}%


%% EUCLIDEAN DIVISION
\newcommand\PolDivide[4]{% #3=quotient, #4=remainder of #1 by #2
    \POL@divide{#1}{#2}%
    \expandafter\let\csname POLuserpol@#3\endcsname\POL@Q
    \POL@newpol{#3}%
    \expandafter\let\csname POLuserpol@#4\endcsname\POL@R
    \POL@newpol{#4}%
}%
\newcommand\PolQuo[3]{% #3=quotient of #1 by #2
    \POL@divide{#1}{#2}%
    \expandafter\let\csname POLuserpol@#3\endcsname\POL@Q
    \POL@newpol{#3}%
}%
\newcommand\PolRem[3]{% #3=remainder of #1 by #2
    \POL@divide{#1}{#2}%
    \expandafter\let\csname POLuserpol@#3\endcsname\POL@R
    \POL@newpol{#3}%
}%
\newcommand\POL@divide[2]{%
    \begingroup
      \let\xintScalarSub\xintSub
      \let\xintScalarAdd\xintAdd
      \let\xintScalarMul\xintMul
      \let\xintScalarDiv\xintDiv
      \expandafter\let\expandafter\POL@A\csname POLuserpol@#1\endcsname
      \expandafter\let\expandafter\POL@B\csname POLuserpol@#2\endcsname
      \POL@div@c
      \let\POL@Q\POL@result
      \ifnum\POL@degQ<\z@
          \let\POL@R\POL@A
      \else
        \count@\numexpr\POL@degR+\@ne\relax
        \POL@resultfromarray R%
        \let\POL@R\POL@result
      \fi
    \expandafter
    \endgroup
    \expandafter
    \def\csname POL@Q\expandafter\expandafter\expandafter\endcsname
        \expandafter\expandafter\expandafter{\expandafter\POL@Q\expandafter}%
        \expandafter
    \def\csname POL@R\expandafter\endcsname\expandafter{\POL@R}%
}%


%% GCD
\newcommand\PolGCD[3]{% sets #3 to the (unitary) G.C.D. of #1 and #2
    \POL@GCD{#1}{#2}{#3}%
    \POL@newpol{#3}%
}%
\def\POL@GCD #1#2#3{%
    \begingroup
      \let\xintScalarSub\xintSub
      \let\xintScalarAdd\xintAdd
      \let\xintScalarMul\xintMul
      \let\xintScalarDiv\xintDiv
      \expandafter\let\expandafter\POL@A\csname POLuserpol@#1\endcsname
      \expandafter\let\expandafter\POL@B\csname POLuserpol@#2\endcsname
      \expandafter\POL@split\POL@A;\POL@degA\POL@polA
      \expandafter\POL@split\POL@B;\POL@degB\POL@polB
      \ifnum\POL@degA<\z@
        \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
      \fi
        {\ifnum\POL@degB<\z@
          \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
        \fi
          {\def\POL@result{-1.\empty{0/1[0]}}}%
          {\xintAssignArray\POL@polB\to\POL@arrayB
           \POL@normalize{B}%
           \POL@gcd@exit BA}}%
        {\ifnum\POL@degB<\z@
          \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
        \fi
          {\xintAssignArray\POL@polA\to\POL@arrayA
           \POL@normalize{A}%
           \POL@gcd@exit AB}%
          {\ifnum\POL@degA<\POL@degB\space
             \let\POL@tmp\POL@B\let\POL@B\POL@A\let\POL@A\POL@tmp
             \let\POL@tmp\POL@degB\let\POL@degB\POL@degA\let\POL@degA\POL@tmp
             \let\POL@tmp\POL@polB\let\POL@polB\POL@polA\let\POL@polA\POL@tmp
           \fi
           \xintAssignArray\POL@polA\to\POL@arrayA
           \xintAssignArray\POL@polB\to\POL@arrayB
           \POL@gcd AB%
           }}%
    \expandafter
    \endgroup
    \expandafter\def\csname POLuserpol@#3\expandafter\endcsname
        \expandafter{\POL@result}%
}%
\def\POL@normalize#1{%
    \expandafter\def\expandafter\POL@tmp\expandafter
        {\csname POL@array#1\csname POL@array#10\endcsname\endcsname}%
    \edef\POL@normalize@leading{\POL@tmp}%
    \expandafter\def\POL@tmp{1/1[0]}%
    \count@\csname POL@deg#1\endcsname\space
    \xintloop
    \ifnum\count@>\z@
      \expandafter\edef\csname POL@array#1\the\count@\endcsname
      {\xintIrr{\xintScalarDiv
                   {\csname POL@array#1\the\count@\endcsname}%
                   {\POL@normalize@leading}}[0]}%
    \advance\count@\m@ne
    \repeat
}%
\def\POL@gcd#1#2{%
    \POL@normalize{#2}%
    \edef\POL@degQ{\the\numexpr\csname POL@deg#1\endcsname
                              -\csname POL@deg#2\endcsname}%
    \count@\numexpr\csname POL@deg#1\endcsname+\@ne\relax
    \count\tw@\numexpr\POL@degQ+\@ne\relax
    \xintloop
      \POL@gcd@getremainder@loopbody#1#2%
    \ifnum\count\tw@>\z@
    \repeat
    \expandafter\def\csname POL@array#10\endcsname{1}%
    \xintloop
    \xintiiifZero{\csname POL@array#1\the\count@\endcsname}%
      {\iftrue}%
      {\iffalse}%
    \advance\count@\m@ne
    \repeat
    \expandafter\edef\csname POL@deg#1\endcsname{\the\numexpr\count@-\@ne}%
    \ifnum\count@<\@ne
      \expandafter\POL@gcd@exit
    \else
      \expandafter\edef\csname POL@array#10\endcsname{\the\count@}%
      \expandafter\POL@gcd
    \fi{#2}{#1}%
}%
\def\POL@gcd@getremainder@loopbody#1#2{%
  \edef\POL@gcd@ratio{\csname POL@array#1\the\count@\endcsname}%
  \advance\count@\m@ne
  \advance\count\tw@\m@ne
  \count4 \count@
  \count6 \csname POL@deg#2\endcsname\space
  \xintloop
  \ifnum\count6>\z@
    \expandafter\edef\csname POL@array#1\the\count4\endcsname
      {\xintScalarSub
          {\csname POL@array#1\the\count4\endcsname}%
          {\xintScalarMul
            {\POL@gcd@ratio}%
            {\csname POL@array#2\the\count6\endcsname}}}%
    \advance\count4 \m@ne
    \advance\count6 \m@ne
  \repeat
}%
\def\POL@gcd@exit#1#2{%
    \count@\numexpr\csname POL@deg#1\endcsname+\@ne\relax
    \POL@resultfromarray #1%
}%


%% TODO: BEZOUT


%% DIFFERENTIATION
\def\POL@diff@loop@one #1/#2[#3]#4%
  {\xintIrr{\xintiiMul{#4}{#1}/#2[0]}[#3]}%
\def\POL@diff#1{\POL@diff@loop1.}%
\def\POL@diff@loop#1.#2{%
  \if\relax#2\expandafter\xint_gob_til_dot\fi
  {\expandafter\POL@diff@loop@one\romannumeral0\xintraw{#2}{#1}}%
  \expandafter\POL@diff@loop\the\numexpr#1+\@ne.%
}%
\newcommand\PolDiff[1][1]{%
   % optional parameter is how many times to derivate
   % first mandatory arg is name of polynomial function to derivate,
   % same name as in \NewPolExpr
   % second mandatory arg name of derivative
   \edef\POL@iterindex{\the\numexpr#1\relax}%
   \ifnum\POL@iterindex<\z@
       \expandafter\@firstoftwo
   \else
       \expandafter\@secondoftwo
   \fi
   {\PolAntiDiff[-\POL@iterindex]}{\POL@Diff}%
}%
\def\POL@Diff{%
   \ifcase\POL@iterindex\space
      \expandafter\POL@Diff@no
   \or\expandafter\POL@Diff@one
   \else\xint_afterfi{\POL@Iterate\POL@Diff@one}%
   \fi
}%
\def\POL@Diff@no #1#2{\POL@let{#2}{#1}}%
\def\POL@Diff@one #1#2{\POL@Diff@@one {#1}{#2}\POL@newpol{#2}}%
\def\POL@Diff@@one#1#2{%
   \expandafter\expandafter\expandafter\POL@split
      \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
   \ifnum\POL@var@deg<\@ne
      \@namedef{POLuserpol@#2}{-1.\empty{0/1[0]}}%
   \else
     \edef\POL@var@coeffs{\expandafter\POL@diff\POL@var@coeffs\relax}%
     \expandafter\edef\csname POLuserpol@#2\endcsname
        {\the\numexpr\POL@var@deg-\@ne.\noexpand\empty\POL@var@coeffs}%
   \fi
}%
% lazy way but allows to share with AntiDiff
\def\POL@Iterate#1#2#3{%
    \begingroup
    \xintverbosefalse
        #1{#2}{#3}%
        \xintloop
        \ifnum\POL@iterindex>\tw@
          #1{#3}{#3}%
          \edef\POL@iterindex{\the\numexpr\POL@iterindex-\@ne}%
        \repeat
   \expandafter
   \endgroup\expandafter
   \def\csname POLuserpol@#3\expandafter\endcsname
       \expandafter{\romannumeral`^^@\csname POLuserpol@#3\endcsname}%
   #1{#3}{#3}%
}%


%% ANTI-DIFFERENTIATION
\def\POL@antidiff@loop@one #1/#2[#3]#4%
  {\xintIrr{#1/\xintiiMul{#4}{#2}[0]}[#3]}%
\def\POL@antidiff{\POL@antidiff@loop1.}%
\def\POL@antidiff@loop#1.#2{%
  \if\relax#2\expandafter\xint_gob_til_dot\fi
  {\expandafter\POL@antidiff@loop@one\romannumeral0\xintraw{#2}{#1}}%
  \expandafter\POL@antidiff@loop\the\numexpr#1+\@ne.%
}%
\newcommand\PolAntiDiff[1][1]{%
   % optional parameter is how many times to derivate
   % first mandatory arg is name of polynomial function to derivate,
   % same name as in \NewPolExpr
   % second mandatory arg name of derivative
   \edef\POL@iterindex{\the\numexpr#1\relax}%
   \ifnum\POL@iterindex<\z@
       \expandafter\@firstoftwo
   \else
       \expandafter\@secondoftwo
   \fi
   {\PolDiff[-\POL@iterindex]}{\POL@AntiDiff}%
}%
\def\POL@AntiDiff{%
   \ifcase\POL@iterindex\space
      \expandafter\POL@AntiDiff@no
   \or\expandafter\POL@AntiDiff@one
   \else\xint_afterfi{\POL@Iterate\POL@AntiDiff@one}%
   \fi
}%
\let\POL@AntiDiff@no\POL@Diff@no
\def\POL@AntiDiff@one #1#2{\POL@AntiDiff@@one{#1}{#2}\POL@newpol{#2}}%
\def\POL@AntiDiff@@one#1#2{%
   \expandafter\expandafter\expandafter\POL@split
      \csname POLuserpol@#1\endcsname;\POL@var@deg\POL@var@coeffs
   \ifnum\POL@var@deg<\z@
      \@namedef{POLuserpol@#2}{-1.\empty{0/1[0]}}%
   \else
     \edef\POL@var@coeffs{\expandafter\POL@antidiff\POL@var@coeffs\relax}%
     \expandafter\edef\csname POLuserpol@#2\endcsname
         {\the\numexpr\POL@var@deg+\@ne.\noexpand\empty{0/1[0]}\POL@var@coeffs}%
   \fi
}%

%% IContent and \PolMakePrimitive (0.5)
\def\POL@aux@mgcd@loop#1#2{%
    \if\relax#2\expandafter\POL@aux@mgcd@exit\fi
    \expandafter
    \POL@aux@mgcd@loop\romannumeral0\POL@aux@gcd#1.#2.%
}%
\def\POL@aux@mgcd@exit
    \expandafter
    \POL@aux@mgcd@loop\romannumeral0\POL@aux@gcd#1.\relax.{\xintiiabs{#1}}%
\def\POL@aux@gcd#1.#2.{%
    \if0\xintiiSgn{#1}\expandafter\POL@aux@gcd@exit\fi
    \expandafter\POL@aux@gcd\romannumeral0\xintmod {#2}{#1}.#1.}%
\def\POL@aux@gcd@exit
    \expandafter\POL@aux@gcd\romannumeral0\xintmod #1#2.#3.{{#1}}%

\def\POL@icontent #1{\romannumeral0\expandafter
    \POL@aux@mgcd@loop\romannumeral`^^@#1\relax}%

\newcommand\PolIContent[1]{\romannumeral0\expandafter
    \POL@aux@mgcd@loop\romannumeral`^^@\PolToList{#1}\relax}%


\def\POL@makeprim@macro#1%
   {\xintREZ{\xintNum{\xintDiv{#1}{\POL@makeprim@icontent}}}}%
\newcommand\PolMakePrimitive[1]{%
    % This does not need a full user declared polynomial on input, only
    % a \POLuserpol@name macro, but on output it is fully declared
    \edef\POL@makeprim@icontent{\PolIContent{#1}}%
    \PolMapCoeffs\POL@makeprim@macro{#1}%
}%
\def\POL@makeprimitive#1{%
    % Avoids declaring the polynomial, internal usage in \PolToSturm
    \edef\POL@makeprim@icontent{\PolIContent{#1}}%
    \POL@mapcoeffs\POL@makeprim@macro{#1}%
}%


%% Sturm Algorithm (polexpr 0.4)
%% 0.5 uses primitive polynomials for faster evaluations afterwards
%% 0.6 corrects misuse of \@ifstar! (mumble). \PolToSturm* was broken.
%% 0.6's \PolToSturm* defines both normalized and unnormalized, the
%%       unnormalized using two underscores, so both are available
%%       Sole difference is that \PolToSturm* also declares them as 
%%       user polynomials, whereas the non-starred only keeps the macros
%%       holding the coefficients in memory
%% 0.6 fixes the case of a constant polynomial P which caused division
%%     by zero error from P'.
\newcommand\PolToSturm{\@ifstar{\PolToSturm@@}{\PolToSturm@}}%
\def\POL@aux@toint#1{\xintREZ{\xintNum{#1}}}% for polynomials with int. coeffs!
%% Attention that some macros rely upon this one setting \POL@sturmname
%% and \POL@sturm@N as it does
\def\PolToSturm@#1#2{%
  \edef\POL@sturmname{#2}%
  % 0.6 uses 2 underscores (one before index, one after) to keep in memory
  % the unnormalized chain
  % This supposes #1 to be a genuine polynomial, not only a name with
  % a \POLuserpol@#1 macro
  \POL@let{\POL@sturmname _0_}{#1}%
  \ifnum\PolDegree{#1}=\z@
     \def\POL@sturm@N{0}%
     \POL@count\z@
     % if I applied the same as for positive degree, I should make it -1
     % if constant is negative. I also don't worry if polynomial is zero.
     \@namedef{POLuserpol@\POL@sturmname _0}{0.\empty{1/1[0]}}%
  \else
     \ifPOL@tosturm@makefirstprimitive\POL@makeprimitive{\POL@sturmname _0_}\fi
     \POL@tosturm@dosturm
  \fi
  \expandafter
  \let\csname PolSturmChainLength_\POL@sturmname\endcsname\POL@sturm@N
  % declare the normalized ones as full-fledged polynomials
  % \POL@count\z@
  \xintloop
    \POL@newpol{\POL@sturmname _\the\POL@count}%
  \unless\ifnum\POL@sturm@N=\POL@count
    \advance\POL@count\@ne
  \repeat
}%
\def\PolToSturm@@#1#2{\PolToSturm@{#1}{#2}\POL@tosturm@declareunnormalized}%
\def\POL@tosturm@declareunnormalized{%
  % optionally declare also the unnormalized ones
  \POL@count\z@
  \xintloop
      \POL@newpol{\POL@sturmname _\the\POL@count _}%
  \unless\ifnum\POL@sturm@N=\POL@count
      \advance\POL@count\@ne
  \repeat
}%
\def\POL@tosturm@dosturm{%
  \POL@Diff@@one{\POL@sturmname _0_}{\POL@sturmname _1_}%
  % re-utiliser \POL@varcoeffs directement?
  \POL@makeprimitive{\POL@sturmname _1_}% does not do \POL@newpol
  \POL@count\@ne
  \xintloop
    \POL@divide{\POL@sturmname _\the\numexpr\POL@count-\@ne\relax _}%
               {\POL@sturmname _\the\POL@count _}%
    \expandafter\POL@split\POL@R;\POL@degR\POL@polR
  \unless\ifnum\POL@degR=\m@ne
    \advance\POL@count\@ne
    \expandafter\let
    \csname POLuserpol@\POL@sturmname _\the\POL@count _\endcsname\POL@R
    \edef\POL@makeprim@icontent{-\POL@icontent\POL@polR}%
    % this avoids the \POL@newpol from \PolMapCoeffs
    \POL@mapcoeffs\POL@makeprim@macro{\POL@sturmname _\the\POL@count _}%
  \repeat
  \edef\POL@sturm@N{\the\POL@count}%
  % normalize (now always done even by starred variant)
  \ifnum\PolDegree{\POL@sturmname _\POL@sturm@N _}>\z@
    % \POL@count\POL@sturm@N\relax
    \xintloop
      \advance\POL@count\m@ne
      \POL@divide{\POL@sturmname _\the\POL@count _}%
                 {\POL@sturmname _\POL@sturm@N _}%
      \expandafter
       \let\csname POLuserpol@\POL@sturmname _\the\POL@count\endcsname\POL@Q
      % quotient actually belongs to Z[X] and is primitive
      \POL@mapcoeffs\POL@aux@toint{\POL@sturmname _\the\POL@count}%    
    \ifnum\POL@count>\z@
    \repeat
    \@namedef{POLuserpol@\POL@sturmname _\POL@sturm@N}{0.\empty{1/1[0]}}%
  \else % they are already normalized
    \advance\POL@count\@ne % attention to include last one also
    \xintloop
      \advance\POL@count\m@ne
      \expandafter\let
       \csname POLuserpol@\POL@sturmname _\the\POL@count\expandafter\endcsname
       \csname POLuserpol@\POL@sturmname _\the\POL@count _\endcsname
    \ifnum\POL@count>\z@
    \repeat
  \fi
  % Back to \PolToSturm@, \POL@count holds 0
}%
\newcommand\PolSturmChainLength[1]
    {\romannumeral`^^@\csname PolSturmChainLength_#1\endcsname}%

\newcommand\PolSetToSturmChainSignChangesAt[4][\global]{%
  \edef\POL@sturmchain@X{\xintREZ{#4}}%
  \edef\POL@sturmname{#3}%
  \edef\POL@sturmlength{\PolSturmChainLength{\POL@sturmname}}%
  \POL@sturmchain@getSV@at\POL@sturmchain@X
  #1\let#2\POL@sturmchain@SV
}%
\def\POL@sturmchain@getSV@at#1{% ATTENTION USES \POL@count
  \def\POL@sturmchain@SV{0}%
  \edef\POL@sturmchain@sign{\xintiiSgn{\POL@eval{\POL@sturmname _0}{#1}}}%
  \let\POL@isolz@lastsign\POL@sturmchain@sign
  \POL@count \z@
  \ifnum\POL@isolz@lastsign=\z@
    \edef\POL@isolz@lastsign
        {\xintiiSgn{\POL@eval{\POL@sturmname _1}{#1}}}%
    \POL@count \@ne
  \fi
  \xintloop
  \unless\ifnum\POL@sturmlength=\POL@count
  \advance\POL@count \@ne
    \edef\POL@isolz@newsign
      {\xintiiSgn{\POL@eval{\POL@sturmname _\the\POL@count}{#1}}}%
    \ifnum\POL@isolz@newsign=\numexpr-\POL@isolz@lastsign\relax
      \edef\POL@sturmchain@SV{\the\numexpr\POL@sturmchain@SV+\@ne}%
      \let\POL@isolz@lastsign=\POL@isolz@newsign
    \fi
  \repeat
}%
\newcommand\PolSetToNbOfZerosWithin[5][\global]{%
  \edef\POL@tmpA{\xintREZ{#4}}%
  \edef\POL@tmpB{\xintREZ{#5}}%
  \edef\POL@sturmname{#3}%
  \edef\POL@sturmlength{\PolSturmChainLength{\POL@sturmname}}%
  \POL@sturmchain@getSV@at\POL@tmpA
  \let\POL@SVA\POL@sturmchain@SV
  \POL@sturmchain@getSV@at\POL@tmpB
  \let\POL@SVB\POL@sturmchain@SV
  \ifnum\POL@SVA<\POL@SVB\space
    #1\edef#2{\the\numexpr\POL@SVB-\POL@SVA}%
  \else
    #1\edef#2{\the\numexpr\POL@SVA-\POL@SVB}%
  \fi
}%


% 0.6 added starred variant to count multiplicities
% 0.7 added double starred variant to locate all rational roots
\newcommand\PolSturmIsolateZeros{\@ifstar
      {\PolSturmIsolateZerosAndGetMultiplicities}%
      {\PolSturmIsolateZeros@}%
}%
\newcommand\PolSturmIsolateZerosAndGetMultiplicities{\@ifstar
      {\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots}%
      {\PolSturmIsolateZerosAndGetMultiplicities@}%
}%
% on aurait besoin de ça dans xint, mais il aurait un \xintRaw{#1} alors
\def\POL@xintfrac@getNDE #1%
   {\expandafter\POL@xintfrac@getNDE@i\romannumeral`^^@#1}%
\def\POL@xintfrac@getNDE@i #1/#2[#3]#4#5#6{\def#4{#1}\def#5{#2}\def#6{#3}}%
\newcommand\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots[2][\empty]{%
  \PolSturmIsolateZerosAndFindRationalRoots[#1]{#2}%
  \ifnum\POL@isolz@NbOfRoots>\z@
     % get multiplicities of irrational (real) roots, if any
     \ifnum\POL@findrat@nbofirrroots>\z@
        \POL@findrat@getirrmult
     \fi
     \POL@isolzmult@defvar@M
  \fi
}%
% added at 0.7
\newcommand\PolSturmIsolateZerosAndFindRationalRoots[2][\empty]{%
  % #1 optional E such that roots are searched in -10^E < x < 10^E
  % both -10^E and +10^E must not be roots!
  % #2 name of Sturm chain (already pre-computed)
  \edef\POL@sturmname{#2}%
  \edef\POL@sturm@N{\@nameuse{PolSturmChainLength_\POL@sturmname}}%
  % isolate the roots (detects case of constant polynomial)
  \PolSturmIsolateZeros@{\POL@sturmname}%
  \ifnum\POL@isolz@NbOfRoots=\z@
     % no real roots, define empty arrays nevertheless
     \begingroup\globaldefs\@ne
     \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL@sturmname*\endcsname
     \expandafter\xintAssignArray\expandafter\to\csname POL_RI\POL@sturmname*\endcsname
     \endgroup
  \else
     % all we currently know is that multiplicities are at least one
     \begingroup\globaldefs\@ne
     \expandafter\POL@initarray\csname POL_ZM\POL@sturmname*\endcsname{1}%
     \endgroup
     % on ne va pas utiliser de Horner, mais des divisions par X - x, et ces
     % choses vont évoluer, ainsi que le coefficient dominant entier
     % (pour \POL@divide entre autres if faut des noms de user pol)
     \expandafter\let
         \csname POLuserpol@\POL@sturmname\POL@sqfnorr\expandafter\endcsname
         \csname POLuserpol@\POL@sturmname _0\endcsname
     \expandafter\let
         \csname POLuserpol@\POL@sturmname\POL@norr\expandafter\endcsname
         \csname POLuserpol@\POL@sturmname _0_\endcsname
     % attention formé avec\xintREZ d'où le \xintAbs pas \xintiiAbs
     % D and its exponent E will get updated along the way
     \edef\POL@findrat@D{\xintAbs{\PolLeadingCoeff{\POL@sturmname _0}}}%
     \POL@xintfrac@getNDE\POL@findrat@D\POL@findrat@Dint\POL@_\POL@findrat@Dexp
     \xintiiifOne{\POL@findrat@Dint}
        {\let\POL@findrat@E\POL@findrat@Dexp} % aussi ok pour 1[0]
        {\edef\POL@findrat@E{\the\numexpr\xintLen{\POL@findrat@Dint}%
                                                 +\POL@findrat@Dexp}}%
% ATTENTION QUE LA CONVENTION DE SIGNE POUR \POL@findrat@E EST OPPOSÉE À CELLE
% POUR LE CODE PLUS ANCIEN FAISANT "REFINE"
     \POL@initarray\POL@IfMultIsKnown\xint_secondoftwo
     \let\POL@findrat@nbofirrroots\POL@isolz@NbOfRoots
     % find all rational roots, and their multiplicities,
     % factor them out in passing from original (Sturm root) polynomial
     \ifnum\POL@findrat@E<7
         \def\POL@findrat@index{1}%
         \POL@findrat@loop@secondpass@direct
     \else
     % we do a first pass scanning for "small" roots p/q (i.e. q < 1000)
         \def\POL@findrat@index{1}%
         \POL@findrat@loop@firstpass
     % and now we do the final pass finding them all
         \def\POL@findrat@index{1}%
         \POL@findrat@loop@secondpass
     \fi
     % declare the new polynomials
     \POL@newpol{\POL@sturmname\POL@sqfnorr}%  without multiplicities
     \POL@newpol{\POL@sturmname\POL@norr}% with multiplicities
     % declare the array holding the interval indices for the rational roots
     \expandafter\POL@findrat@doRRarray\csname POL_RI\POL@sturmname*\endcsname
  \fi
}%
\def\POL@findrat@doRRarray#1{%
  % il faudrait un \xintAssignArray* qui fasse même expansion que \xintFor*
  \edef\POL@temp{%
    \xintiloop[1+1]
    \romannumeral0\csname POL_ZK\POL@sturmname*\xintiloopindex\endcsname
    \xintbracediloopindex % I should have named it \xintiloopbracedindex...
    {}%
    \ifnum\xintiloopindex<\POL@isolz@NbOfRoots\space
    \repeat }%
  \begingroup\globaldefs\@ne
  % attention de ne surtout pas faire un \expandafter ici, car en cas d'un
  % seul item, \xintAssignArray l'unbraces...
  \xintAssignArray\POL@temp\to#1%
  \endgroup
}%
\def\POL@findrat@loop@firstpass{%
  \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}%
     \POL@findrat@loop@decimal% get its multiplicity
     \POL@findrat@loop@aa     % refine interval and check
  \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}%
  \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots
  \else
    \expandafter\POL@findrat@loop@firstpass
  \fi
}%
\def\POL@findrat@loop@aa{%
  % we do a first pass to identify roots with denominators < 1000
  \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-6}%
  % attention that perhaps now the root is known!
  \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}%
      \POL@findrat@loop@decimal
      \POL@findrat@loop@a
}%
\def\POL@findrat@loop@decimal{% we have an already found decimal root
  % we do not go via @storeit, as it is already stored
  % j'ai beaucoup hésité néanmoins, car je pourrais faire \xintIrr ici,
  % mais attention aussi à l'interaction avec le \PolDecToString. Les racines
  % trouvées directement (qui peuvent être des nombres décimaux) sont elles
  % stockées comme fraction irréductibles (modulo action additionnelle de 
  % \PolDecToString). 
  \POL@xintfrac@getNDE
    {\xintIrr{\POL@xintexprGetVar{\POL@sturmname L_\POL@findrat@index}}[0]}%
    \POL@findrat@xN\POL@findrat@xD\POl@_
  % we can't move this to updatequotients because other branch will
  % need to do the division first anyhow
  \edef\POLuserpol@_findrat@oneterm{1.\noexpand\empty
        {\xintiiOpp\POL@findrat@xN/1[0]}{\POL@findrat@xD/1[0]}}%
  \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult.
  %\expandafter\POL@split\POL@R;\POL@degR\POL@polR
  \POL@findrat@loop@updatequotients
  \POL@findrat@loop@getmultiplicity
}%
% lacking from xint 1.3c, but \xintSgn has overhead, so we define ii version
\def\xintiiifNeg{\romannumeral0\xintiiifneg }%
\def\xintiiifneg #1%
{%
    \ifcase \xintiiSgn{#1}
               \expandafter\xint_stop_atsecondoftwo
            \or\expandafter\xint_stop_atsecondoftwo
          \else\expandafter\xint_stop_atfirstoftwo
    \fi
}%
\def\POL@findrat@getE #1/1[#2]{#2}% /1 as it should be there.
% so an error will arise if not but cf \POL@refine@getE where I did not put it
\def\POL@findrat@loop@a{%
  % attention that the width may have been already smaller than 10^{-6}
  \POL@get@IsoLeft@rawin
  \POL@get@IsoRight@rawin
  \edef\POL@findrat@localW
      {\the\numexpr-\expandafter\POL@findrat@getE
  % do I really need the \xintREZ?
       \romannumeral0\xintrez
         {\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}%
      }% at least 6, maybe larger
  \expandafter\POL@get@Int@aux
              \POL@IsoLeft@rawin\POL@IsoLeft@Int{-\POL@findrat@localW}%
  \expandafter\POL@get@Int@aux
              \POL@IsoRight@rawin\POL@IsoRight@Int{-\POL@findrat@localW}%
  % in case of odd, some waste here
  \edef\POL@findrat@halflocalW{\the\numexpr(\POL@findrat@localW+1)/2-1}%
  % Legendre Theorem will be used now but we separate a branch where
  % everything can be done with \numexpr
  \ifnum\POL@findrat@localW>9
    % not implemented yet by lazyness!
    % this root will be handled in second pass only
  \else
    \POL@findrat@gcdloop
  \fi
}%
\def\POL@findrat@gcdloop{%
  % we must be careful with sign
  % but we are certain no extremity is a root
  \let\POL@findrat@ifnegative\xint_secondoftwo
  \xintiiifSgn\POL@IsoLeft@Int
    \POL@findrat@gcdloop@n
    \POL@error@thisisimpossible
    \POL@findrat@gcdloop@p
}%
\def\POL@findrat@gcdloop@n{%
  \let\POL@findrat@ifnegative\xint_firstoftwo
  \let\POL@temp\POL@IsoRight@Int
  \edef\POL@IsoRight@Int{\xintiiOpp{\POL@IsoLeft@Int}}%
  \edef\POL@IsoLeft@Int{\xintiiOpp{\POL@temp}}%
  \POL@findrat@gcdloop@p
}%
\def\POL@findrat@gcdloop@p{%
  \edef\POL@findrat@gcdloop@Ap{\xintDec{\xintDouble\POL@IsoRight@Int}}%
  \edef\POL@findrat@gcdloop@A
  % at most 2e9: this is acceptable to \numexpr
      {2\romannumeral\xintreplicate\POL@findrat@localW{0}}%
  \xintAssign
    \xintiiDivision\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A
  \to\POL@findrat@gcdloop@B\POL@findrat@gcdloop@An
  % on fait de la tambouille pour n'utiliser que \numexpr par la suite
  % le reste @An est < 2.10^9 au pire donc ok pour \numexpr
  % we will drop integral part in our updating P
  \let\POL@findrat@gcdloop@Binitial\POL@findrat@gcdloop@B
  \def\POL@findrat@gcdloop@B{0}%  do as if B1 = 0
  \def\POL@findrat@gcdloop@Pp{1}% P0
  \def\POL@findrat@gcdloop@P{0}%  P1
  \def\POL@findrat@gcdloop@Qp{0}% Q0
  \def\POL@findrat@gcdloop@Q{1}%  Q1
  % A2=An can not be zero, as Ap (=A0) is odd and A (=A1=200...000) is even
  % first Binitial + P1/Q1 ( = Binitial) can not be root
  \let\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A % A1
  \let\POL@findrat@gcdloop@A\POL@findrat@gcdloop@An % A2
  \def\next{\POL@findrat@gcdloop@update}%
  \def\POL@findrat@gcdloop@done{0}%
  \POL@findrat@gcdloop@body
}%
\def\POL@findrat@gcdloop@body{%
    % annoying that \numexpr has no divmod... use counts? but groups annoying
    \edef\POL@findrat@gcdloop@B
        {\the\numexpr(\POL@findrat@gcdloop@Ap+\POL@findrat@gcdloop@A/2)/%
                      \POL@findrat@gcdloop@A - \@ne}%
    \edef\POL@findrat@gcdloop@An
        {\the\numexpr\POL@findrat@gcdloop@Ap-%
                     \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@A}%
    \edef\POL@findrat@gcdloop@Pn
        {\the\numexpr\POL@findrat@gcdloop@Pp+%
                     \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@P}%
    \edef\POL@findrat@gcdloop@Qn
        {\the\numexpr\POL@findrat@gcdloop@Qp+%
                     \POL@findrat@gcdloop@B*\POL@findrat@gcdloop@Q}%
    \ifnum\expandafter\xintLength\expandafter{\POL@findrat@gcdloop@Qn}%
          >\POL@findrat@halflocalW\space
       \let\next\empty % no solution was found
    \else
    % with these conditions on denom, only candidates are by Legendre
    % theorem among the convergents as computed here
       \ifnum\POL@findrat@gcdloop@Qn>\POL@findrat@gcdloop@An\space
         % means that P/Q is in interval and is thus a candidate
         % it is automatically irreducible
         \edef\POL@findrat@x{\xintiiAdd
           {\xintiiMul{\POL@findrat@gcdloop@Qn}{\POL@findrat@gcdloop@Binitial}}%
           {\POL@findrat@gcdloop@Pn}/\POL@findrat@gcdloop@Qn[0]}%
         \POL@findrat@gcdloop@testit
         \if1\POL@findrat@gcdloop@done
            \let\next\empty % a solution was found
         \fi
       \fi
    \fi
    \next
}%
\def\POL@findrat@gcdloop@update{%
    \ifnum\POL@findrat@gcdloop@An>\z@
        \let\POL@findrat@gcdloop@Ap\POL@findrat@gcdloop@A
        \let\POL@findrat@gcdloop@A\POL@findrat@gcdloop@An
        \let\POL@findrat@gcdloop@Pp\POL@findrat@gcdloop@P
        \let\POL@findrat@gcdloop@P\POL@findrat@gcdloop@Pn
        \let\POL@findrat@gcdloop@Qp\POL@findrat@gcdloop@Q
        \let\POL@findrat@gcdloop@Q\POL@findrat@gcdloop@Qn
    \expandafter\POL@findrat@gcdloop@body
    \fi
}%
\def\POL@findrat@gcdloop@testit{%
  % zero should never occur here
  \POL@findrat@ifnegative{\edef\POL@findrat@x{-\POL@findrat@x}}{}%
  \POL@xintfrac@getNDE\POL@findrat@x\POL@findrat@xN\POL@findrat@xD\POL@_
  \edef\POLuserpol@_findrat@oneterm{1.\noexpand\empty
      {\xintiiOpp{\POL@findrat@xN}/1[0]}{\POL@findrat@xD/1[0]}}%
  \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult.
  \expandafter\POL@split\POL@R;\POL@degR\POL@polR
  \ifnum\POL@degR=\m@ne % found a root
    \POL@findrat@loop@storeit
    \POL@findrat@loop@updatequotients
    \POL@findrat@loop@getmultiplicity % will continue updating the mult. one
    \def\POL@findrat@gcdloop@done{1}%
  \fi
}%            
% This is second phase
\def\POL@findrat@loop@secondpass{%
  \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}%
     {}% nothing more to be done, already stored
     \POL@findrat@loop@bb % refine interval and check
  \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}%
  \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots
  \else
     \expandafter\POL@findrat@loop@secondpass
  \fi
}%
\def\POL@findrat@loop@secondpass@direct{%
  \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}%
     \POL@findrat@loop@decimal
     \POL@findrat@loop@bb
  \edef\POL@findrat@index{\the\numexpr\POL@findrat@index+\@ne}%
  \ifnum\POL@findrat@index>\POL@isolz@NbOfRoots
  \else
     \expandafter\POL@findrat@loop@secondpass@direct
  \fi
}%
\def\POL@findrat@loop@bb{%
  \PolEnsureIntervalLength{\POL@sturmname}{\POL@findrat@index}{-\POL@findrat@E}%
  % ATTENTION THAT PERHAPS NOW THE ROOT IS KNOWN!
  \PolSturmIfZeroExactlyKnown{\POL@sturmname}{\POL@findrat@index}%
      \POL@findrat@loop@decimal
      \POL@findrat@loop@b
}%
\def\POL@findrat@loop@b{%
  \edef\POL@findrat@Lscaled{\xintMul{\POL@findrat@D}%
       {\POL@xintexprGetVar{\POL@sturmname L_\POL@findrat@index}}}%
  \edef\POL@findrat@Rscaled{\xintMul{\POL@findrat@D}%
       {\POL@xintexprGetVar{\POL@sturmname R_\POL@findrat@index}}}%
  \xintiiifNeg{\POL@findrat@Lscaled}% using ii version is an abuse
    {% negative interval (right bound possibly zero!)
     % truncate towards zero (i.e. to the right) the left bound
     \edef\POL@findrat@Num{\xintNum{\POL@findrat@Lscaled}/1[0]}%
     % interval boundaries are not root hence in case that was exact
     % this will not be found as a root; check if in interval
     \xintifLt\POL@findrat@Num\POL@findrat@Rscaled
         \POL@findrat@loop@c
         {}% iterate
    }%
    {% positive interval (left bound possibly zero!)
     % truncate towards zero (i.e. to the left) the right bound
     \edef\POL@findrat@Num{\xintNum{\POL@findrat@Rscaled}/1[0]}%
     % check if in interval
     \xintifGt\POL@findrat@Num\POL@findrat@Lscaled
         \POL@findrat@loop@c
         {}% iterate
    }%
}%
\def\POL@findrat@loop@c{%
  % safer to do the edef as \POL@findrat@x used later in storeit
  \edef\POL@findrat@x{\xintIrr{\xintDiv\POL@findrat@Num\POL@findrat@D}[0]}%
  \POL@xintfrac@getNDE\POL@findrat@x\POL@findrat@xN\POL@findrat@xD\POL@_
  \edef\POLuserpol@_findrat@oneterm{1.\noexpand\empty
      {\xintiiOpp{\POL@findrat@xN}/1[0]}{\POL@findrat@xD/1[0]}}%
  \POL@divide{\POL@sturmname\POL@sqfnorr}{_findrat@oneterm}% the one without mult.
  \expandafter\POL@split\POL@R;\POL@degR\POL@polR
  \ifnum\POL@degR=\m@ne % found a root
    \POL@findrat@loop@storeit
    \POL@findrat@loop@updatequotients
    \POL@findrat@loop@getmultiplicity % will continue updating the mult. one
  \fi
  % iterate
}%
\def\POL@findrat@loop@storeit{%
  % update storage, I can not use storeleftandright here (due to rawout etc...)
  \expandafter
  \xdef\csname POL_ZL\POL@sturmname*\POL@findrat@index\endcsname
                                     {\PolDecToString{\POL@findrat@x}}%
  \global\expandafter
  \let\csname POL_ZR\POL@sturmname*\POL@findrat@index\expandafter\endcsname
       \csname POL_ZL\POL@sturmname*\POL@findrat@index\endcsname
  \global\expandafter
  \let\csname POL_ZK\POL@sturmname*\POL@findrat@index\endcsname
        \xint_stop_atfirstoftwo
  \begingroup\xintglobaldefstrue
    % skip some overhead of \xintdefvar...
    \XINT_expr_defvar_one{\POL@sturmname L_\POL@findrat@index}%
       {\csname .=\POL@findrat@x\endcsname}%
    \XINT_expr_defvar_one{\POL@sturmname R_\POL@findrat@index}%
       {\csname .=\POL@findrat@x\endcsname}%
    \XINT_expr_defvar_one{\POL@sturmname Z_\POL@findrat@index _isknown}%
       {\csname .=1\endcsname}%
  \endgroup
}%
\def\POL@findrat@loop@updatequotients{%
  % attention last division must have been one testing vanishing of\POL@sqfnorr
  \expandafter\let\csname POLuserpol@\POL@sturmname\POL@sqfnorr\endcsname\POL@Q
  % quotient belongs to Z[X] and is primitive
  \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@sqfnorr}%
  % update the one with multiplicities
  \POL@divide{\POL@sturmname\POL@norr}{_findrat@oneterm}%
  \expandafter\let\csname POLuserpol@\POL@sturmname\POL@norr\endcsname\POL@Q
  \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@norr}
  % updating of \POL@findrat@D at end of execution of getmultiplicity
}%
\def\POL@findrat@loop@getmultiplicity{%
  % the one without multiplicity must not be divided again!
  % check if we have remaining multiplicity
  \POL@divide{\POL@sturmname\POL@norr}{_findrat@oneterm}%
  \expandafter\POL@split\POL@R;\POL@degR\POL@polR
  \ifnum\POL@degR=\m@ne % yes
    \expandafter\let\csname POLuserpol@\POL@sturmname\POL@norr\endcsname\POL@Q
    \POL@mapcoeffs\POL@aux@toint{\POL@sturmname\POL@norr}%
    \expandafter
    \xdef
    \csname POL_ZM\POL@sturmname*\POL@findrat@index\endcsname
      {\the\numexpr
       \csname POL_ZM\POL@sturmname*\POL@findrat@index\endcsname+\@ne}%
    \expandafter\POL@findrat@loop@getmultiplicity
  \else 
  % done with multiplicity for this rational root, update stuff
    \edef\POL@findrat@nbofirrroots
         {\the\numexpr\POL@findrat@nbofirrroots-\@ne}%
    \@namedef{POL@IfMultIsKnown\POL@findrat@index}{\xint_firstoftwo}%
    \edef\POL@findrat@D{\xintAbs{\PolLeadingCoeff{\POL@sturmname\POL@sqfnorr}}}%
    \POL@xintfrac@getNDE\POL@findrat@D\POL@findrat@Dint\POL@_\POL@findrat@Dexp
    \xintiiifOne{\POL@findrat@Dint}
        {\let\POL@findrat@E\POL@findrat@Dexp} % aussi ok pour 1[0]
        {\edef\POL@findrat@E{\the\numexpr\xintLen{\POL@findrat@Dint}%
                                                 +\POL@findrat@Dexp}}%
  \fi
}%
\def\POL@findrat@getirrmult{%
  % first get the GCD of remaining pol with its derivative
  \POL@divide{\POL@sturmname\POL@norr}{\POL@sturmname\POL@sqfnorr}%
  \expandafter\let
    % attention au _ (cf. grosse astuce pour \POL@isolzmult@loop)
    \csname POLuserpol@@_1\POL@sturmname _\endcsname\POL@Q
  \ifnum\PolDegree{@_1\POL@sturmname _}>\z@
    % il reste des multiplicités (mais peut-être pour des racines complexes)
    % (ou pour des racines en-dehors de l'intervalle optionnel)
    % attention recyclage ici de \POL@isolzmult@loop qui dépend de
    % la  grosse astuce avec \@gobble
    \POL@makeprimitive{@_1\POL@sturmname _}%
    \let\POL@originalsturmname\POL@sturmname
    % trick to get isolzmult@loop to define @@lastGCD to @_1sturmname_
    % because it will do \POL@sturmname _\POL@sturm@N _
    \edef\POL@sturmname{@_1\POL@sturmname}%
    \let\POL@sturm@N\@gobble% !
    \let\POL@isolz@NbOfRoots@with_unknown_mult\POL@findrat@nbofirrroots
    \POL@tosturm@makefirstprimitivefalse
    \POL@isolzmult@loop
    \POL@tosturm@makefirstprimitivetrue
    \let\POL@sturmname\POL@originalsturmname
  \fi
}%


\newcommand\PolSturmIsolateZerosAndGetMultiplicities@[2][\empty]{%
  % #1 optional E such that roots are searched in -10^E < x < 10^E
  % both -10^E and +10^E must not be roots!
  % #2 name of Sturm chain (already pre-computed)
  \edef\POL@sturmname{#2}%
  \edef\POL@sturm@N{\@nameuse{PolSturmChainLength_\POL@sturmname}}%
  % isolate the roots (detects case of constant polynomial)
  \PolSturmIsolateZeros@{\POL@sturmname}%
  \ifnum\POL@isolz@NbOfRoots=\z@
  % no roots, define empty array nevertheless
     \begingroup\globaldefs\@ne
     \expandafter\xintAssignArray\expandafter\to\csname POL_ZM\POL@sturmname*\endcsname
     \endgroup
  \else
     % all we currently know is that multiplicities are at least one
     \begingroup\globaldefs\@ne
     \expandafter\POL@initarray\csname POL_ZM\POL@sturmname*\endcsname{1}%
     \endgroup
     % check if GCD had positive degree (hence some roots, maybe complex, have
     % multiplicity)
     \ifnum\PolDegree{\POL@sturmname _\POL@sturm@N _}>\z@
         % scratch array of flags to signal known multiplicities
         \POL@initarray\POL@IfMultIsKnown\xint_secondoftwo
         % this count has utility for the case there are other roots
         % either complex or outside interval (in case of optional argument)
         \let\POL@isolz@NbOfRoots@with_unknown_mult\POL@isolz@NbOfRoots
         % store Sturm chain name, it is needed and altered in isolzmult@loop
         \let\POL@originalsturmname\POL@sturmname
         \POL@tosturm@makefirstprimitivefalse
         \POL@isolzmult@loop
         \POL@tosturm@makefirstprimitivetrue
         \let\POL@sturmname\POL@originalsturmname
     \fi
     \POL@isolzmult@defvar@M
  \fi
}%
\def\POL@isolzmult@defvar@M{%
  % Attention that is used not only in ...GetMultiplicities@ but also
  % in FindRationalRoots
  \begingroup\xintglobaldefstrue
  % added at 0.7
  \let\x\POL@isolz@NbOfRoots
    \xintloop
    % skip some overhead of \xintdefvar...
      \XINT_expr_defvar_one{\POL@sturmname M_\x}%
         {\csname .=\csname POL_ZM\POL@sturmname*\x\endcsname\endcsname}%
    \edef\x{\the\numexpr\x-\@ne}%
    \ifnum\x>\z@
    \repeat
  \endgroup
}%
\def\POL@isolzmult@loop{%
   % we are here only if last iteration gave a new GCD still of degree > 0
   % \POL@sturm@N is the one from last iteration
   % Attention to not use \POL@sturmname directly in first arg. of \PolToSturm
   % Attention that we need for the case of known roots also to have the last
   % GCD (with its multiplicities) known as a genuine polynomial
   % - because of usage of \POL@eval in @isknown branch
   % - because \PolToSturm@ does a \POL@let which would be anomalous
   %   if the extended structure is not existing
   \edef\POL@isolzmult@lastGCD{\POL@sturmname _\POL@sturm@N _}%
   \edef\POL@isolzmult@newsturmname{@_1\POL@sturmname}%
   \POL@newpol{\POL@isolzmult@lastGCD}%
   \PolToSturm@{\POL@isolzmult@lastGCD}{\POL@isolzmult@newsturmname}%
   % now both \POL@sturmname and \POL@sturm@N have changed
   \edef\POL@isolzmult@newGCDdegree{\PolDegree{\POL@sturmname _\POL@sturm@N _}}%
   \let\POL@isolzmult@index\POL@isolz@NbOfRoots
   \xintloop
     % ATTENTION that this executes macros which also modifies \POL@sturmname!
     % (but not \POL@sturm@N)
     \POL@isolzmult@doone
     \edef\POL@isolzmult@index{\the\numexpr\POL@isolzmult@index-\@ne}%
   \if1\ifnum\POL@isolz@NbOfRoots@with_unknown_mult=\z@ 0\fi
       \ifnum\POL@isolzmult@index=\z@ 0\fi 1%
   \repeat
   \let\POL@sturmname\POL@isolzmult@newsturmname
   \if1\ifnum\POL@isolz@NbOfRoots@with_unknown_mult=\z@ 0\fi
       % (if new GCD is constant, time to abort)
       \ifnum\POL@isolzmult@newGCDdegree=\z@ 0\fi 1%
     \expandafter\POL@isolzmult@loop
   \fi
}%
\def\POL@isolzmult@doone{%
   \csname POL@IfMultIsKnown\POL@isolzmult@index\endcsname
     {}% nothing to do
     {\POL@SturmIfZeroExactlyKnown{\POL@originalsturmname}%
                                  {\POL@isolzmult@index}%
           \POL@isolzmult@loop@isknown
           \POL@isolzmult@loop@isnotknown
      \POL@isolzmult@loop@sharedbody
     }%
}%
\def\POL@isolzmult@loop@isknown{%
   \xintifZero
   % attention that \POL@eval requires a declared polynomial
     {\POL@eval{\POL@isolzmult@lastGCD}%
               {\POL@xintexprGetVar{\POL@originalsturmname L_\POL@isolzmult@index}}}%
   {\let\POL@isolzmult@haszero\@ne}%
   {\let\POL@isolzmult@haszero\z@}%
}%
\def\POL@isolzmult@loop@isnotknown{%
   \edef\POL@isolzmult@loop@A 
        {\POL@xintexprGetVar{\POL@originalsturmname L_\POL@isolzmult@index}}
   \edef\POL@isolzmult@loop@B
        {\POL@xintexprGetVar{\POL@originalsturmname
            R_\POL@isolzmult@index}}
   % attention that \PolSetToNbOfZerosWithin sets \POL@sturmname to 2nd argument
   \PolSetToNbOfZerosWithin
      \POL@isolzmult@haszero  % nb of zeros A < x <= B, here 0 or 1
      \POL@isolzmult@newsturmname
      \POL@isolzmult@loop@A
      \POL@isolzmult@loop@B
}%
\def\POL@isolzmult@loop@sharedbody{%
   \ifnum\POL@isolzmult@haszero>\z@
     \expandafter
     \xdef
     \csname POL_ZM\POL@originalsturmname*\POL@isolzmult@index\endcsname
       {\the\numexpr
        \csname POL_ZM\POL@originalsturmname
                            *\POL@isolzmult@index\endcsname+\@ne}%
   \else
     % multiplicity now known, no need to check this index in future
     \@namedef{POL@IfMultIsKnown\POL@isolzmult@index}{\xint_firstoftwo}%
     \edef\POL@isolz@NbOfRoots@with_unknown_mult
       {\the\numexpr\POL@isolz@NbOfRoots@with_unknown_mult-\@ne}%
   \fi
}%


\newcommand\PolSturmIsolateZeros@[2][\empty]{%
  % #1 optional E such that roots are searched in -10^E < x < 10^E
  % both -10^E and +10^E must not be roots!
  % #2 name of Sturm chain (already pre-computed from a given polynomial)
  % For reasons I have forgotten (no time now) this code **must** be used
  % with a *normalized* Sturm chain.
  \edef\POL@sturmname{#2}%
  \edef\POL@sturmlength{\PolSturmChainLength{#2}}%
  % attention to constant polynomial, we must redefine the arrays then
  \ifnum\POL@sturmlength>\z@
   \ifx\empty#1\relax
    \POL@isolz@getsignchanges@plusinf
    \POL@isolz@getsignchanges@minusinf
   \else
    \edef\POL@isolz@E{\the\numexpr\xint_zapspaces #1 \xint_gobble_i\relax}%
    \POL@sturmchain@getSV@at{1[\POL@isolz@E]}%
    \let\POL@isolz@plusinf@SV  \POL@sturmchain@SV
    \let\POL@isolz@plusinf@sign\POL@sturmchain@sign
    \POL@sturmchain@getSV@at{-1[\POL@isolz@E]}%
    \let\POL@isolz@minusinf@SV  \POL@sturmchain@SV
    \let\POL@isolz@minusinf@sign\POL@sturmchain@sign
    \ifnum\POL@isolz@plusinf@sign=\z@
        \PackageError{polexpr}%
{The polynomial #2 vanishes at set upper bound 10^\POL@isolz@E}%
{Compile again with a bigger exponent in source. (X to abort).}%
    \fi
    \ifnum\POL@isolz@minusinf@sign=\z@
        \PackageError{polexpr}%
{The polynomial #2 vanishes at set lower bound -10^\POL@isolz@E}%
{Compile again with a bigger exponent in source. (X to abort).}%
    \fi
   \fi
   \edef\POL@isolz@NbOfRoots
      {\the\numexpr\POL@isolz@minusinf@SV-\POL@isolz@plusinf@SV}%
  \else
    % constant polynomial
   \def\POL@isolz@NbOfRoots{0}%
  \fi
  \ifnum\POL@isolz@NbOfRoots=\z@
     \begingroup\globaldefs\@ne
     \expandafter\xintAssignArray\expandafter\to\csname POL_ZL#2*\endcsname
     \expandafter\xintAssignArray\expandafter\to\csname POL_ZR#2*\endcsname
     \expandafter\xintAssignArray\expandafter\to\csname POL_ZK#2*\endcsname
     \endgroup
  \else
     \begingroup\globaldefs\@ne
     \expandafter\POL@initarray\csname POL_ZL#2*\endcsname{0}%
     \expandafter\POL@initarray\csname POL_ZR#2*\endcsname{0}%
     \expandafter\POL@initarray\csname POL_ZK#2*\endcsname
                 \xint_stop_atsecondoftwo
     \endgroup
     \ifx\empty#1\relax\expandafter\POL@isolz@getaprioribound\fi
     \expandafter\POL@isolz@main
  \fi
}%
\def\POL@initarray#1#2{%
% ATTENTION, if only one item, \xintAssignArray UNBRACES IT
% so we use an \empty trick to avoid that. Maybe considered a bug of xinttools?
  \expandafter\xintAssignArray\expandafter\empty
    \romannumeral\xintreplicate{\POL@isolz@NbOfRoots}{{#2}}\to#1%
}%
\def\POL@isolz@getsignchanges@plusinf{%
  % Count number of sign changes at plus infinity in Sturm sequence
  \def\POL@isolz@plusinf@SV{0}%
  \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _0}}}%
  \let\POL@isolz@plusinf@sign\POL@isolz@lastsign
  \POL@count\@ne
  \xintloop
    \edef\POL@isolz@newsign
        {\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _\the\POL@count}}}%
    \unless\ifnum\POL@isolz@newsign=\POL@isolz@lastsign
       \edef\POL@isolz@plusinf@SV{\the\numexpr\POL@isolz@plusinf@SV+\@ne}%
    \fi
    \let\POL@isolz@lastsign=\POL@isolz@newsign
  \ifnum\POL@sturmlength>\POL@count
    \advance\POL@count\@ne
  \repeat
}%
\def\POL@isolz@getsignchanges@minusinf{%
  % Count number of sign changes at minus infinity in Sturm sequence
  \def\POL@isolz@minusinf@SV{0}%
  \edef\POL@isolz@lastsign{\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _0}}}%
  \ifodd\PolDegree{\POL@sturmname _0}
      \edef\POL@isolz@lastsign{\xintiiOpp{\POL@isolz@lastsign}}%
  \fi
  \let\POL@isolz@minusinf@sign\POL@isolz@lastsign
  \POL@count\@ne
  \xintloop
    \edef\POL@isolz@newsign
      {\xintiiSgn{\PolLeadingCoeff{\POL@sturmname _\the\POL@count}}}%
    \ifodd\PolDegree{\POL@sturmname _\the\POL@count}
      \edef\POL@isolz@newsign{\xintiiOpp{\POL@isolz@newsign}}%
    \fi
    \unless\ifnum\POL@isolz@newsign=\POL@isolz@lastsign
       \edef\POL@isolz@minusinf@SV{\the\numexpr\POL@isolz@minusinf@SV+\@ne}%
    \fi
    \let\POL@isolz@lastsign=\POL@isolz@newsign
  \ifnum\POL@sturmlength>\POL@count
    \advance\POL@count\@ne
  \repeat
}%
% utility macro for a priori bound on root decimal exponent, via Float Rounding
\def\POL@isolz@updateE #1e#2;%
  {\unless\ifnum#2<\POL@isolz@E\space\edef\POL@isolz@E{\the\numexpr#2+\@ne}\fi}%
\def\POL@isolz@getaprioribound{%
  \PolAssign{\POL@sturmname _0}\toarray\POL@arrayA
  \edef\POL@isolz@leading{\POL@arrayA{\POL@arrayA{0}}}%
  \POL@count\z@
  \xintloop
  \advance\POL@count\@ne
  \ifnum\POL@arrayA{0}>\POL@count
     \expandafter\edef\csname POL@arrayA\the\POL@count\endcsname
        {\xintDiv{\POL@arrayA\POL@count}\POL@isolz@leading}%
  \repeat
  \def\POL@isolz@E{1}% WE SEEK SMALLEST E SUCH HAT -10^E < roots < +10^E
  \advance\POL@count\m@ne
  \xintloop
  \ifnum\POL@count>\z@
     \expandafter\POL@isolz@updateE
     % use floating point to get decimal exponent
     \romannumeral0\xintfloat[4]% should I use with [2] rather? (should work)
       {\xintAdd{1/1[0]}{\xintAbs{\POL@arrayA\POL@count}}};%
  \advance\POL@count\m@ne
  \repeat
  % \ifxintverbose\xintMessage{polexpr}{Info}%
  %   {Roots a priori bounded in absolute value by 10 to the \POL@isolz@E.}%
  % \fi
}%
\def\POL@IsoRight@raw{\POL@IsoRight@Int/1[\POL@isolz@E]}%
\def\POL@IsoLeft@raw {\POL@IsoLeft@Int/1[\POL@isolz@E]}%
\def\POL@IsoRight@rawout{%
    \ifnum\POL@IsoRightSign=\z@\expandafter\xintREZ\fi\POL@IsoRight@raw
}%
\def\POL@IsoLeft@rawout{%
    \ifnum\POL@IsoRightSign=\z@
       \expandafter\xint_firstoftwo\else\expandafter\xint_secondoftwo
    \fi{\xintREZ\POL@IsoRight@raw}%
       {\POL@IsoLeft@Int/1[\POL@isolz@E]}%
}%
\def\POL@isolz@main {%
% NOTE 2018/02/16. THIS WILL PRESUMABLY BE RE-ORGANIZED IN FUTURE TO DO
% FIRST POSITIVE ROOTS THEN NEGATIVE ROOTS VIA CHANGE OF VARIABLE TO OPPOSITE.
  \global\POL@isolz@nextwillneedrefinefalse
  \def\POL@IsoRight@Int{0}%
  \POL@sturmchain@getSV@at\POL@IsoRight@raw
  \let\POL@IsoRightSV  \POL@sturmchain@SV
  \let\POL@IsoRightSign\POL@sturmchain@sign
  \let\POL@IsoAtZeroSV  \POL@IsoRightSV
  \let\POL@IsoAtZeroSign\POL@IsoRightSign
  \ifnum\POL@IsoAtZeroSign=\z@
    \xdef\POL@isolz@IntervalIndex
      {\the\numexpr\POL@isolz@minusinf@SV-\POL@IsoRightSV}%
    \POL@refine@storeleftandright % store zero root, \POL@IsoRightSign is zero
    \edef\POL@IsoRightSV{\the\numexpr\POL@IsoRightSV+\@ne}%
% subtlety here if original polynomial had multiplicities, but ok. I checked!
    \edef\POL@IsoRightSign % evaluated twice, but that's not so bad
      {\xintiiOpp{\xintiiSgn{\POL@eval{\POL@sturmname _1}{0/1[0]}}}}%
  \fi
  \def\POL@IsoLeft@Int{-1}% -10^E isn't a root!
  \let\POL@IsoLeftSV  \POL@isolz@minusinf@SV
  \let\POL@IsoLeftSign\POL@isolz@minusinf@sign
  % \POL@IsoRight@SV was modified if zero is a root
  \edef\POL@isolz@NbOfNegRoots{\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV}%
  \gdef\POL@isolz@IntervalIndex{0}%
  \let\POL@isolz@@E\POL@isolz@E
  \ifnum\POL@isolz@NbOfNegRoots>\z@
% refactored at 0.7 to fix cases leading to an intervals with zero as end-point
    \POL@isolz@findroots@neg
  \fi
  \let\POL@isolz@E\POL@isolz@@E
  \def\POL@IsoLeft@Int{0}%
  \let\POL@IsoLeftSV  \POL@IsoAtZeroSV  % véritable SV en zéro
  \let\POL@IsoLeftSign\POL@IsoAtZeroSign% véritable signe en zéro
  \ifnum\POL@IsoLeftSign=\z@
    \xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}%
  \fi
  \let\POL@@IsoRightSV  \POL@isolz@plusinf@SV
  \let\POL@@IsoRightSign\POL@isolz@plusinf@sign % 10^E not a root!
  \edef\POL@isolz@NbOfPosRoots
    {\the\numexpr\POL@IsoLeftSV-\POL@@IsoRightSV}% attention @@
  \ifnum\POL@isolz@NbOfPosRoots>\z@
    % always do that to avoid zero as end-point whether it is a root or not
    \global\POL@isolz@nextwillneedrefinetrue
    \POL@isolz@findroots@pos
  \fi
}%
\def\POL@isolz@findroots@neg{%
  \def\POL@IsoRight@Int{-1}%
  \POL@isolz@findnextzeroboundeddecade@neg
  \def\POL@IsoLeft@Int{-10}%
  \let\POL@@IsoRightSign\POL@IsoRightSign % a zero there is possible
  \let\POL@@IsoRightSV  \POL@IsoRightSV
  % this will do possibly recursive \POL@isolz@check's
  \POL@isolz@explorenexteightsubdecades@neg
  \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfNegRoots\space
    % above did not explore -2, -1 for this optimization (SV known at Right)
    \def\POL@IsoRight@Int{-1}%
    \let\POL@IsoRightSign\POL@@IsoRightSign
    \let\POL@IsoRightSV  \POL@@IsoRightSV
    \POL@isolz@check
    \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfNegRoots\space
      \def\POL@IsoLeft@Int{-1}%
      \let\POL@IsoLeftSign\POL@@IsoRightSign
      \let\POL@IsoLeftSV  \POL@@IsoRightSV
      % I don't like being inside TeX conditionals
      \expandafter\expandafter\expandafter\POL@isolz@findroots@neg
    \fi
  \fi
}%
\def\POL@isolz@findnextzeroboundeddecade@neg{%
  \xintloop
    \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
    \POL@sturmchain@getSV@at\POL@IsoRight@raw
    \let\POL@IsoRightSV  \POL@sturmchain@SV
    \let\POL@IsoRightSign\POL@sturmchain@sign
    % would an \ifx test be quicker? (to be checked)
  \ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space
  % no roots in-between, iterate
  \repeat
}%
\def\POL@isolz@explorenexteightsubdecades@neg{%
  \xintloop
    \edef\POL@IsoRight@Int{\the\numexpr\POL@IsoLeft@Int+\@ne}%
     % we could arguably do a more efficient dichotomy here
    \POL@sturmchain@getSV@at\POL@IsoRight@raw
    \let\POL@IsoRightSV  \POL@sturmchain@SV
    \let\POL@IsoRightSign\POL@sturmchain@sign
    \POL@isolz@check % may recurse if multiple roots are to be found
    \ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfNegRoots\space
      \expandafter\xintbreakloop
    \fi
    \let\POL@IsoLeft@Int\POL@IsoRight@Int
    \let\POL@IsoLeftSign\POL@IsoRightSign
    \let\POL@IsoLeftSV\POL@IsoRightSV
  \ifnum\POL@IsoRight@Int < -\tw@
  \repeat
}%
\def\POL@isolz@findroots@pos{%
  % remark (2018/12/08), this needs some refactoring, I hardly understand
  % the logic and it hides most into the recursion done by \POL@isolz@check
  % It would probably make more sense to proceed like done for the negative
  % but here finding the largest roots first.
  \def\POL@IsoRight@Int{1}%
  \POL@isolz@findnextzeroboundeddecade@pos
  \unless\ifnum\POL@IsoRightSV=\POL@IsoLeftSV\space
    % this actually explores the whole of some interval (0, 10^{e-1}]
    % in a context where some roots are known to be in (10^{e-1}, 10^{e}]
    % and none are larger
    \POL@isolz@check % will recurse inside groups if needed with modified E
  \fi
  % we know get the roots in the last 9 decades from 10^{e-1} to 10^{e}
  % we should arguably do a more efficient dichotomy here
  \def\POL@IsoLeft@Int{1}%
  \let\POL@IsoLeftSV\POL@IsoRightSV
  \let\POL@IsoLeftSign\POL@IsoRightSign
  \xintloop
    \edef\POL@IsoRight@Int{\the\numexpr\POL@IsoLeft@Int+\@ne}%
    \POL@sturmchain@getSV@at\POL@IsoRight@raw
    \let\POL@IsoRightSV  \POL@sturmchain@SV
    \let\POL@IsoRightSign\POL@sturmchain@sign
    \POL@isolz@check % recurses in needed
    \let\POL@IsoLeft@Int\POL@IsoRight@Int
    \let\POL@IsoLeftSign\POL@IsoRightSign
    \let\POL@IsoLeftSV\POL@IsoRightSV
    \ifnum\POL@isolz@IntervalIndex=\POL@isolz@NbOfRoots\space
      \expandafter\xintbreakloop
    \fi
    \ifnum\POL@IsoLeft@Int < \xint_c_ix
  \repeat
  \ifnum\POL@isolz@IntervalIndex<\POL@isolz@NbOfRoots\space
  % get now the last, rightmost, root (or roots)
    \def\POL@IsoRight@Int{10}%
    \let\POL@IsoRightSign\POL@@IsoRightSign
    \let\POL@IsoRightSV\POL@@IsoRightSV
    \POL@isolz@check
  \fi
}%
\def\POL@isolz@findnextzeroboundeddecade@pos{%
  \xintloop
    \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
    \POL@sturmchain@getSV@at\POL@IsoRight@raw
    \let\POL@IsoRightSV  \POL@sturmchain@SV
    \let\POL@IsoRightSign\POL@sturmchain@sign
  \ifnum\POL@IsoRightSV=\POL@@IsoRightSV\space
    \let\POL@@IsoRightSign\POL@IsoRightSign % root here possible!
  \repeat
}%
\def\POL@isolz@check{% \POL@IsoRightSign must be ready for use here
% \ifxintverbose
%   \xintMessage{polexpr}{Info}%
%   {\the\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax\space roots
%    in (\POL@IsoLeft@raw,\POL@IsoRight@raw] (E = \POL@isolz@E)}%
% \fi
    \ifcase\numexpr\POL@IsoLeftSV-\POL@IsoRightSV\relax
    % no root in ]left, right]
      \global\POL@isolz@nextwillneedrefinefalse
    \or
    % exactly one root in ]left, right]
     \xdef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex+\@ne}%
     \ifnum\POL@IsoRightSign=\z@
       % if right boundary is a root, ignore previous flag
       \global\POL@isolz@nextwillneedrefinefalse
     \fi
     % if left boundary is known to have been a root we refine interval
     \ifPOL@isolz@nextwillneedrefine
       \expandafter\expandafter\expandafter\POL@isolz@refine
     \else
       % \POL@IsoRightSign is zero iff root now exactly known
       \POL@refine@storeleftandright
       \ifnum\POL@IsoRightSign=\z@
        \global\POL@isolz@nextwillneedrefinetrue
       \fi
     \fi
    \else
    % more than one root, we need to recurse
     \expandafter\POL@isolz@recursedeeper
    \fi
}%
\def\POL@isolz@recursedeeper{%
% NOTE 2018/02/16. I SHOULD DO A REAL BINARY DICHOTOMY HERE WHICH ON AVERAGE
% SHOULD BRING SOME GAIN (LIKE WHAT IS ALREADY DONE FOR THE "refine" MACROS.
% THUS IN FUTURE THIS MIGHT BE REFACTORED.
\begingroup
  \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
  \edef\POL@@IsoRight@Int{\xintDSL{\POL@IsoRight@Int}}%
  \let\POL@@IsoRightSign  \POL@IsoRightSign
  \let\POL@@IsoRightSV    \POL@IsoRightSV
  \edef\POL@IsoLeft@Int  {\xintDSL{\POL@IsoLeft@Int}}%
  \xintiloop[1+1]
    \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
    \POL@sturmchain@getSV@at\POL@IsoRight@raw
    \let\POL@IsoRightSV  \POL@sturmchain@SV
    \let\POL@IsoRightSign\POL@sturmchain@sign
    \POL@isolz@check
    \let\POL@IsoLeft@Int\POL@IsoRight@Int
    \let\POL@IsoLeftSV\POL@IsoRightSV
    \let\POL@IsoLeftSign\POL@IsoRightSign% not used, actually
  \ifnum\POL@IsoLeftSV=\POL@@IsoRightSV\space
     \expandafter\xintbreakiloop
  \fi
  \ifnum\xintiloopindex < \xint_c_ix
  \repeat
  \let\POL@IsoRight@Int\POL@@IsoRight@Int
  \let\POL@IsoRightSign\POL@@IsoRightSign
  \let\POL@IsoRightSV  \POL@@IsoRightSV
   % if we exited the loop via breakiloop this is superfluous
   % but it only costs one \ifnum
  \POL@isolz@check
\endgroup
}%
\def\POL@isolz@refine{%
  % starting point is first root = left < unique second root < right
  % even if we hit exactly via refinement second root, we set flag false as
  % processing will continue with original right end-point, which isn't a root
  \global\POL@isolz@nextwillneedrefinefalse
\begingroup
  \let\POL@@IsoRightSign\POL@IsoRightSign % already evaluated
  \xintloop
    \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
    \edef\POL@IsoLeft@Int {\xintDSL{\POL@IsoLeft@Int}}%
    \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
    \edef\POL@IsoRightSign
        {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
  \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
  \repeat
  % now second root has been separated from the one at left end point
% we update the storage of the root at left for it to have the same number
% of digits in mantissa. No, I decided not to do that to avoid complications.
  % \begingroup
  %   \let\POL@IsoRight@Int\POL@IsoLeft@Int
  %   \def\POL@IsoRightSign{0}%
  %   \edef\POL@isolz@IntervalIndex{\the\numexpr\POL@isolz@IntervalIndex-\@ne}%
  %   \POL@refine@storeleftandright
  % \endgroup
  \edef\POL@@IsoRight@Int{\xintDSL{\xintInc{\xintDSR{\POL@IsoLeft@Int}}}}%
  \let\POL@IsoLeft@Int\POL@IsoRight@Int
  \let\POL@IsoLeftSign\POL@IsoRightSign
  \ifnum\POL@IsoRightSign=\z@ % check if new Left is actually a root
  \else
    \edef\POL@IsoRight@Int{\xintDec{\POL@@IsoRight@Int}}%
    \edef\POL@IsoRightSign
      {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
    \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
      \POL@refine@doonce % we need to locate in interval (1, 9) in local scale
    \else
    \let\POL@IsoLeft@Int\POL@IsoRight@Int
    \ifnum\POL@IsoRightSign=\z@
      \def\POL@IsoLeftSign{0}%
    \else
      \let\POL@IsoRight@Int\POL@@IsoRight@Int
      % the IsoRightSign is now wrong but here we don't care
    \fi\fi
  \fi
  % on exit, exact root found iff \POL@IsoRightSign is zero
  \POL@refine@storeleftandright
\endgroup
}%
\def\POL@refine@doonce{% if exact root is found, always in IsoRight on exit
% NOTE: FUTURE REFACTORING WILL GET RID OF \xintiiAdd WHICH ARE A BIT COSTLY
% BUT BASICALLY NEEDED TO HANDLE BOTH NEGATIVE AND POSITIVE HERE.
% I WILL RE-ORGANIZE THE WHOLE THING IN FUTURE TO GET ROOTS STARTING FROM
% THE ORIGIN AND SIMPLY RE-LABEL THE NEGATIVE ONE AT THE END. 2018/02/16.
  \let\POL@@IsoRight@Int\POL@IsoRight@Int % 9
  \let\POL@@IsoRightSign\POL@IsoRightSign
  \edef\POL@IsoRight@Int{\xintiiAdd{4}{\POL@IsoLeft@Int}}% 5
  \edef\POL@IsoRightSign
      {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
  \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
    \let\POL@IsoLeft@Int\POL@IsoRight@Int % 5
    \edef\POL@IsoRight@Int{\xintiiAdd{2}{\POL@IsoLeft@Int}}%
    \edef\POL@IsoRightSign
        {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
    \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
      \let\POL@IsoLeft@Int\POL@IsoRight@Int % 7
      \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
      \edef\POL@IsoRightSign
          {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
      \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
        \let\POL@IsoLeft@Int\POL@IsoRight@Int % 8
        \let\POL@IsoRight@Int\POL@@IsoRight@Int % 9
        \let\POL@IsoRightSign\POL@@IsoRightSign % opposite of one at left
      \fi % else 7, 8 with possible root at 8
    \else
    \ifnum\POL@IsoRightSign=\z@
      \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 7
      \def\POL@IsoLeftSign{0}%
    \else
      \let\POL@@IsoRight@Int\POL@IsoRight@Int % 7
      \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 6
      \edef\POL@IsoRightSign
          {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
      \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
          \let\POL@IsoLeft@Int\POL@IsoRight@Int   % 6
          \let\POL@IsoRight@Int\POL@@IsoRight@Int % 7
          \let\POL@IsoRightSign\POL@@IsoRightSign
      \fi % else 5, 6 with possible root at 6
    \fi\fi
  \else
  \ifnum\POL@IsoRightSign=\z@
    \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 5
    \def\POL@IsoLeftSign{0}%
  \else
    \let\POL@@IsoRight@Int\POL@IsoRight@Int % 5
    \edef\POL@IsoRight@Int{\xintiiAdd{2}{\POL@IsoLeft@Int}}%
    \edef\POL@IsoRightSign
        {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
    \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
      \let\POL@IsoLeft@Int\POL@IsoRight@Int % 3
      \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 4
      \edef\POL@IsoRightSign
          {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
      \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
          \let\POL@IsoLeft@Int\POL@IsoRight@Int   % 4
          \let\POL@IsoRight@Int\POL@@IsoRight@Int % 5
          \let\POL@IsoRightSign\POL@@IsoRightSign
      \fi % else 3, 4 with possible root at 4
    \else
    \ifnum\POL@IsoRightSign=\z@
      \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 3
      \def\POL@IsoLeftSign{0}%
    \else
      \let\POL@@IsoRight@Int\POL@IsoRight@Int % 3
      \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}% 2
      \edef\POL@IsoRightSign
          {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
      \ifnum\POL@IsoRightSign=\POL@IsoLeftSign\space
          \let\POL@IsoLeft@Int\POL@IsoRight@Int   % 2
          \let\POL@IsoRight@Int\POL@@IsoRight@Int % 3
          \let\POL@IsoRightSign\POL@@IsoRightSign
      \fi % else 1, 2 with possible root at 2
    \fi\fi
  \fi\fi
}%
\def\POL@refine@storeleftandright{%
    \expandafter
    \xdef\csname POL_ZL\POL@sturmname*\POL@isolz@IntervalIndex\endcsname
         {\PolDecToString{\POL@IsoLeft@rawout}}%
    \expandafter
    \xdef\csname POL_ZR\POL@sturmname*\POL@isolz@IntervalIndex\endcsname
         {\PolDecToString{\POL@IsoRight@rawout}}%
    % added at 0.6
    \ifnum\POL@IsoRightSign=\z@
      \global
      \expandafter
      \let\csname POL_ZK\POL@sturmname*\POL@isolz@IntervalIndex\endcsname
          \xint_stop_atfirstoftwo
    \fi
    \begingroup\xintglobaldefstrue
    % skip some overhead of \xintdefvar...
      \XINT_expr_defvar_one{\POL@sturmname L_\POL@isolz@IntervalIndex}%
         {\csname .=\POL@IsoLeft@rawout\endcsname}%
      \XINT_expr_defvar_one{\POL@sturmname R_\POL@isolz@IntervalIndex}%
         {\csname .=\POL@IsoRight@rawout\endcsname}%
      % added at 0.7
      \XINT_expr_defvar_one{\POL@sturmname Z_\POL@isolz@IntervalIndex _isknown}%
         {\csname .=\ifnum\POL@IsoRightSign=\z@ 1\else 0\fi\endcsname}%
    \endgroup
}%
%% \PolRefineInterval
\def\POL@xintexprGetVar#1{\expandafter\expandafter\expandafter
    \XINT_expr_unlock\csname XINT_expr_var_#1\endcsname}%
% attention, also used by \POL@findrat@loop@a
\def\POL@get@IsoLeft@rawin{%
    \edef\POL@IsoLeft@rawin
     {\POL@xintexprGetVar{\POL@sturmname L_\POL@isolz@IntervalIndex}}%
}%
% attention, also used by \POL@findrat@loop@a
\def\POL@get@IsoRight@rawin{%
    \edef\POL@IsoRight@rawin
     {\POL@xintexprGetVar{\POL@sturmname R_\POL@isolz@IntervalIndex}}%
}%
% attention, also used by \POL@findrat@loop@a
\def\POL@get@Int@aux #1/1[#2]#3#4{\edef#3{\xintDSH{#4-#2}{#1}}}%
\def\POL@get@IsoLeft@Int{%
    \expandafter\POL@get@Int@aux\POL@IsoLeft@rawin\POL@IsoLeft@Int\POL@isolz@E
}%
\newcommand\PolRefineInterval{\@ifstar\POL@srefine@start\POL@refine@start}%
\newcommand\POL@refine@start[3][1]{%
  \edef\POL@isolz@IntervalIndex{\the\numexpr#3}%
  \edef\POL@sturmname{#2}%
  \expandafter\POL@refine@sharedbody\expandafter
    {\expandafter\POL@refine@loop\expandafter{\the\numexpr#1}}%
}%
\def\POL@srefine@start#1#2{%
  \edef\POL@isolz@IntervalIndex{\the\numexpr#2}%
  \edef\POL@sturmname{#1}%
  \POL@refine@sharedbody
    {\let\POL@refine@left@next\POL@refine@main  % we want to recurse if needed
     \let\POL@refine@right@next\POL@refine@main % we want to recurse if needed
     \POL@refine@main}%
}%
\def\POL@refine@sharedbody#1{%
  \POL@get@IsoLeft@rawin
  \edef\POL@IsoLeftSign
      {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoLeft@rawin}}}%
  \ifnum\POL@IsoLeftSign=\z@
  % do nothing if that interval was already a singleton
  \else
  % else both end-points are not roots and there is a single one in-between
    \POL@get@IsoRight@rawin
    \edef\POL@IsoRightSign{\the\numexpr-\POL@IsoLeftSign}%
    \edef\POL@isolz@E{\expandafter\POL@refine@getE
         % je pense que le xintrez ici est superflu
         \romannumeral0\xintrez{\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}}%
    \POL@get@IsoLeft@Int
    \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
    #1%
    \POL@refine@storeleftandright % \POL@IsoRightSign not zero
  \fi
}%
\def\POL@refine@loop#1{%
    \let\POL@refine@left@next \@empty % no recursion at end sub-intervals
    \let\POL@refine@right@next\@empty
    \xintiloop[1+1]
      \POL@refine@main
      \ifnum\POL@IsoRightSign=\z@
        \expandafter\xintbreakiloop
      \fi
    \ifnum\xintiloopindex<#1
    \repeat
}%
\def\POL@refine@main{%
  \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
  \edef\POL@IsoLeft@Int{\xintDSL{\POL@IsoLeft@Int}}%
  \edef\POL@IsoRight@Int{\xintDSL{\POL@IsoRight@Int}}%
  \let\POL@@IsoRight@Int\POL@IsoRight@Int
  \let\POL@@IsoRightSign\POL@IsoRightSign
  \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
  \edef\POL@IsoRightSign
      {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
  \ifnum\POL@IsoRightSign=\z@
   \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 1
   \def\POL@IsoLeftSign{0}%
   \let\POL@next\@empty
  \else
  \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
    \let\POL@next\POL@refine@left@next % may be \@empty or \POL@refine@main for recursion
    \let\POL@refine@right@next\@empty
  \else
    \let\POL@IsoLeft@Int\POL@IsoRight@Int
    \edef\POL@IsoRight@Int{\xintDec{\POL@@IsoRight@Int}}%
    \edef\POL@IsoRightSign
      {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
    \ifnum\POL@IsoRightSign=\z@
     \let\POL@IsoLeft@Int\POL@IsoRight@Int % root at 9
     \def\POL@IsoLeftSign{0}%
     \let\POL@next\@empty
    \else
     \ifnum\POL@IsoRightSign=\POL@@IsoRightSign\space
      \let\POL@next\POL@refine@doonce
     \else
      \let\POL@IsoLeft@Int\POL@IsoRight@Int
      \let\POL@IsoRight@Int\POL@@IsoRight@Int
      \let\POL@IsoRightSign\POL@@IsoRightSign
      \let\POL@next\POL@refine@right@next
      \let\POL@refine@left@next\@empty
     \fi
    \fi
  \fi\fi
  \POL@next
}%
% lacking pre-defined xintfrac macro here (such as an \xintRawExponent)
\def\POL@refine@getE#1[#2]{#2}% \xintREZ already applied, for safety


\newcommand\PolIntervalWidth[2]{%
% le \xintRez est à cause des E positifs, car trailing zéros explicites
% si je travaillais à partir des variables xintexpr directement ne devrait
% pas être nécessaire, mais trop fragile par rapport à chgt internes possibles
     \romannumeral0\xintrez{\xintSub{\@nameuse{POL_ZR#1*}{#2}}%
                                    {\@nameuse{POL_ZL#1*}{#2}}}
}%


\newcommand\PolEnsureIntervalLengths[2]{% #1 = Sturm chain name,
   % localize roots in intervals of length at most 10^{#2}
   \edef\POL@sturmname{#1}%
   \edef\POL@ensure@targetE{\the\numexpr#2}%
   \edef\POL@nbofroots{\csname POL_ZL\POL@sturmname*0\endcsname}%
   \ifnum\POL@nbofroots>\z@
      \expandafter\POL@ensureintervallengths
   \fi
}%
\def\POL@ensureintervallengths{%
   \POL@count\z@
   % \POL@count used by \POL@sturmchain@getSV@at but latter not used
   \xintloop
     \advance\POL@count\@ne
     \edef\POL@isolz@IntervalIndex{\the\POL@count}%
     \POL@ensure@one
   \ifnum\POL@nbofroots>\POL@count
   \repeat
}%
\newcommand\PolEnsureIntervalLength[3]{% #1 = Sturm chain name,
   % #2 = index of interval
   % localize roots in intervals of length at most 10^{#3}
   \edef\POL@sturmname{#1}%
   \edef\POL@ensure@targetE{\the\numexpr#3}%
   \edef\POL@isolz@IntervalIndex{\the\numexpr#2}%
% peut-être autoriser -1, -2, ... ?
   \ifnum\POL@isolz@IntervalIndex>\z@
% 0.7, add this safeguard but attention means this structure must be in place
   \ifnum\csname POL_ZL\POL@sturmname*0\endcsname>\z@
% je ne fais pas les \expandafter mais je préfèrerai ne pas être à l'intérieur
     \POL@ensure@one
   \fi
   \fi
}%
\def\POL@ensure@one{%
    \POL@get@IsoLeft@rawin
    \POL@get@IsoRight@rawin
    \edef\POL@ensure@delta{\xintREZ{\xintSub{\POL@IsoRight@rawin}{\POL@IsoLeft@rawin}}}%
    \xintiiifZero{\POL@ensure@delta}
      {}
      {\edef\POL@isolz@E{\expandafter\POL@refine@getE\POL@ensure@delta}%
       \POL@get@IsoLeft@Int
       \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
       \ifnum\POL@isolz@E>\POL@ensure@targetE\space
         \edef\POL@IsoLeftSign
          {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoLeft@raw}}}%
          % at start left and right are not roots, and values of opposite signs
         % \edef\POL@IsoRightSign{\the\numexpr-\POL@IsoLeftSign}%
         \xintloop
           \POL@ensure@Eloopbody % decreases E by one at each iteration
          % if separation level is still too coarse we recurse at deeper level
         \ifnum\POL@isolz@E>\POL@ensure@targetE\space
         \repeat
         % will check if right is at a zero, it needs \POL@IsoRightSign set up
         \POL@refine@storeleftandright
       \fi
      }%
}%
\def\POL@ensure@Eloopbody {%
    \edef\POL@isolz@E{\the\numexpr\POL@isolz@E-\@ne}%
    \edef\POL@IsoLeft@Int{\xintDSL{\POL@IsoLeft@Int}}%
    % this will loop at most ten times
    \xintloop
      \edef\POL@IsoRight@Int{\xintInc{\POL@IsoLeft@Int}}%
      \edef\POL@IsoRightSign
          {\xintiiSgn{\POL@eval{\POL@sturmname _0}{\POL@IsoRight@raw}}}%
          % if we have found a zero at right boundary the \ifnum test will fail
          % and we exit the loop
          % else we exit the loop if sign at right boundary is opposite of
          % sign at left boundary (the latter is +1 or -1, never 0)
    % this is a bit wasteful if we go ten times to the right, because
    % we know that there the sign will be opposite, evaluation was superfluous
    \ifnum\POL@IsoLeftSign=\POL@IsoRightSign\space
      \let\POL@IsoLeft@Int\POL@IsoRight@Int
    \repeat
          % check for case when we exited the inner loop because we actually
          % found a zero, then we force exit from the main (E decreasing) loop
    \ifnum\POL@IsoRightSign=\z@
      \expandafter\xintbreakloop
    \fi
}%


\catcode`_ 8
\newcommand\PolPrintIntervals
           {\@ifstar{\PolPrintIntervals@@}{\PolPrintIntervals@}}%
\newcommand\PolPrintIntervals@@{%
 \begingroup
   \def\POL@AfterPrintIntervals{\endgroup}%
   \def\arraystretch{2}%
   \let\PolPrintIntervalsPrintExactZero\POL@@PrintIntervalsPrintExactZero
   \let\PolPrintIntervalsUnknownRoot\POL@@PrintIntervalsUnknownRoot
   \let\PolPrintIntervalsKnownRoot\POL@@PrintIntervalsKnownRoot
   \def\PolPrintIntervalsBeginEnv{\[\begin{array}{cl}}%\]
   \def\PolPrintIntervalsEndEnv{\end{array}\]}%
 \PolPrintIntervals@
}%
\newcommand\PolPrintIntervals@[2][Z]{\POL@PrintIntervals{#1}{#2}}%
\newcommand\POL@PrintIntervals[2]{%
   \def\PolPrintIntervalsTheVar{#1}%
   \def\PolPrintIntervalsTheSturmName{#2}%
   \ifnum\@nameuse{POL_ZL#2*}{0}=\z@
     \PolPrintIntervalsNoRealRoots
   \else
     \gdef\PolPrintIntervalsTheIndex{1}%
     \POL@PrintIntervals@DoDefs
     \begingroup\edef\POL@tmp{\endgroup
       \unexpanded\expandafter{\PolPrintIntervalsBeginEnv}%
       \unexpanded\expandafter{\POL@PrintIntervals@Loop}%
       \unexpanded\expandafter{\PolPrintIntervalsEndEnv}%
     }\POL@tmp
   \fi
   \POL@AfterPrintIntervals
   \def\PolPrintIntervalsTheVar{#1}%
   \def\PolPrintIntervalsTheSturmName{#2}%
}%
\let\POL@AfterPrintIntervals\@empty
\newcommand\PolPrintIntervalsNoRealRoots{}%
\newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}%
\newcommand\PolPrintIntervalsEndEnv{\end{array}\]}%
\newcommand\PolPrintIntervalsKnownRoot{%
  &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
  &=&\PolPrintIntervalsPrintExactZero
}%
\newcommand\PolPrintIntervalsUnknownRoot{%
  \PolPrintIntervalsPrintLeftEndPoint&<&%
  \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
  \PolPrintIntervalsPrintRightEndPoint
}%
\newcommand\PolPrintIntervalsPrintExactZero    {\PolPrintIntervalsTheLeftEndPoint}%
\newcommand\PolPrintIntervalsPrintLeftEndPoint {\PolPrintIntervalsTheLeftEndPoint}%
\newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}%
\newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}%
%
\newcommand\POL@@PrintIntervalsKnownRoot{%
   \PolPrintIntervalsPrintMultiplicity&%
   \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
   \PolPrintIntervalsPrintExactZero
}%
\newcommand\POL@@PrintIntervalsPrintExactZero{%
   \displaystyle
   \xintSignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
}%
\newcommand\POL@@PrintIntervalsUnknownRoot{%
   \PolPrintIntervalsPrintMultiplicity&%
   \xintifSgn{\PolPrintIntervalsTheLeftEndPoint}%
     {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
       {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
        \PolPrintIntervalsPrintRightEndPoint\dots}%
      {0>\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}>%
        \PolPrintIntervalsPrintLeftEndPoint}%
      {\PolErrorThisShouldNotHappenPleaseReportToAuthorA}}%
    {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
      {\PolErrorThisShouldNotHappenPleaseReportToAuthorB}%
      {\PolErrorThisShouldNotHappenPleaseReportToAuthorC}%
      {0<\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}<%
        \PolPrintIntervalsPrintRightEndPoint}}%
    {\xintifSgn{\PolPrintIntervalsTheRightEndPoint}
      {\PolErrorThisShouldNotHappenPleaseReportToAuthorD}%
      {\PolErrorThisShouldNotHappenPleaseReportToAuthorE}%
      {\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}=%
        \PolPrintIntervalsPrintLeftEndPoint\dots}}%
}%
%
\catcode`_ 11
\def\POL@PrintIntervals@Loop{%
  \POL@SturmIfZeroExactlyKnown\PolPrintIntervalsTheSturmName
                              \PolPrintIntervalsTheIndex
    \PolPrintIntervalsKnownRoot
    \PolPrintIntervalsUnknownRoot
  \xdef\PolPrintIntervalsTheIndex{\the\numexpr\PolPrintIntervalsTheIndex+\@ne}%
  \unless\ifnum\PolPrintIntervalsTheIndex>
               \@nameuse{POL_ZL\PolPrintIntervalsTheSturmName*0}
     \POL@PrintIntervals@DoDefs
     \xint_afterfi{\\\POL@PrintIntervals@Loop}%
  \fi
}%
\def\POL@PrintIntervals@DoDefs{%
  \xdef\PolPrintIntervalsTheLeftEndPoint{%
     \csname POL_ZL\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
     \endcsname
  }%
  \xdef\PolPrintIntervalsTheRightEndPoint{%
     \csname POL_ZR\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
     \endcsname
  }%
  \xdef\PolPrintIntervalsTheMultiplicity{%
     \ifcsname POL_ZM\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
     \endcsname
     \csname POL_ZM\PolPrintIntervalsTheSturmName*\PolPrintIntervalsTheIndex
     \endcsname
     \else
     ?% or use 0 ?
     \fi
  }%
}%


\newcommand\PolSturmIfZeroExactlyKnown[2]{% #1 = sturmname, #2=index
    \romannumeral0\csname POL_ZK#1*\endcsname{#2}%
}%
\newcommand\POL@SturmIfZeroExactlyKnown[2]{% #1 = sturmname, #2=index
    \romannumeral0\csname POL_ZK#1*\the\numexpr#2\endcsname
}%
\newcommand\PolSturmIsolatedZeroMultiplicity[2]{%
    \romannumeral`^^@\csname POL_ZM#1*\endcsname{#2}%
}%
\newcommand\PolSturmIsolatedZeroLeft[2]{%
    \romannumeral`^^@\csname POL_ZL#1*\endcsname{#2}%
}%
\newcommand\PolSturmIsolatedZeroRight[2]{%
    \romannumeral`^^@\csname POL_ZR#1*\endcsname{#2}%
}%
\newcommand\PolSturmNbOfIsolatedZeros[1]{%
    \romannumeral`^^@\csname POL_ZL#1*0\endcsname
}%
\newcommand\PolSturmRationalRoot[2]{%
    \romannumeral`^^@\csname POL_ZL#1*%
                     \csname POL_RI#1*\endcsname{#2}\endcsname
}%
\newcommand\PolSturmRationalRootIndex[2]{%
    \romannumeral`^^@\csname POL_RI#1*\endcsname{#2}%
}%
\newcommand\PolSturmRationalRootMultiplicity[2]{%
    \romannumeral`^^@\csname POL_ZM#1%
                     *\csname POL_RI#1*\endcsname{#2}\endcsname
}%
\newcommand\PolSturmNbOfRationalRoots[1]{%
    \romannumeral`^^@\csname POL_RI#1*0\endcsname
}%
\newcommand\PolSturmNbOfRationalRootsWithMultiplicities[1]{%
% means the \POL@norr must not have been changed in-between...
    \the\numexpr\PolDegree{#1}-\PolDegree{#1\POL@norr}\relax
}%


\let\PolDecToString\xintDecToString


\newcommand\PolMakeMonic[1]{%
    \edef\POL@leadingcoeff{\PolLeadingCoeff{#1}}%
    \edef\POL@leadingcoeff@inverse{\xintDiv{1/1[0]}{\POL@leadingcoeff}}%
    \PolMapCoeffs{\xintMul{\POL@leadingcoeff@inverse}}{#1}%
}%


%% CORE ALGEBRA MACROS
%% We do this non-expandably, but in a nestable way... this is the whole
%% point because \xintdeffunc as used by \poldef creates a big nested macro.
%% The idea is to execute it with another meaning given to \xintAdd etc..,
%% so that it operates on "polynomials". This is a mixture of expandable
%% and non-expandable techniques.
\def\POL@get#1#2#3{%
    \global\POL@polfalse
    \begingroup
        \def\POL@result{#3}%
        #3%
        \expandafter
    \endgroup
    \expandafter\def\expandafter#1\expandafter{\POL@result}%
    \unless\ifPOL@pol
        % avoid expanding more than twice #3
        \edef#1{#3}%
        \xintiiifZero{#1}%
            {\def#1{-1.\empty{0/1[0]}}}%
            {\edef#1{0.\noexpand\empty{#1}}}%
    \fi
    #2%
}%
%% ADDITION
\def\POL@add  {\POL@get\POL@A\POL@add@b}%
\def\POL@add@b{\POL@get\POL@B\POL@add@c}%
\def\POL@add@c{%
    \global\POL@poltrue
    \POL@ifZero\POL@A
        {\let\POL@result\POL@B}%
        {\POL@ifZero\POL@B
             {\let\POL@result\POL@A}%
             {\POL@@add}}%
}%
\def\POL@@add{%
  \expandafter\POL@split\POL@A;\POL@degA\POL@polA
  \expandafter\POL@split\POL@B;\POL@degB\POL@polB
  \ifnum\POL@degA>\POL@degB\relax
    \xintAssignArray\POL@polA\to\POL@arrayA
    \xintAssignArray\POL@polB\to\POL@arrayB
  \else
    \xintAssignArray\POL@polB\to\POL@arrayA
    \xintAssignArray\POL@polA\to\POL@arrayB
    \let\POL@tmp\POL@degB\let\POL@degB\POL@degA\let\POL@degA\POL@tmp
  \fi
  \count@\z@
  \xintloop
     \advance\count@\@ne
     \expandafter\edef\csname POL@arrayA\the\count@\endcsname
          {\xintScalarAdd{\@nameuse{POL@arrayA\the\count@}}%
                         {\@nameuse{POL@arrayB\the\count@}}}%
  \unless\ifnum\POL@degB<\count@
  \repeat
  \count@\@nameuse{POL@arrayA0} % 1+\POL@degA
  % trim zero leading coefficients (we could check for equal degrees,
  % but would not bring much as anyhow loop exists immediately if not)
  \xintloop
   % this abuses that \POL@arrayA0 is never zero
   \xintiiifZero{\@nameuse{POL@arrayA\the\count@}}%
      {\iftrue}%
      {\iffalse}%
  \advance\count@\m@ne
  \repeat
  \POL@resultfromarray A% attention that \POL@arrayA0 not updated
}%

%% MULTIPLICATION
\def\POL@mul  {\POL@get\POL@A\POL@mul@b}%
\def\POL@mul@b{\POL@get\POL@B\POL@mul@c}%
\def\POL@mul@c{%
    \global\POL@poltrue
    \POL@ifZero\POL@A
        {\def\POL@result{-1.\empty{0/1[0]}}}%
        {\POL@ifZero\POL@B
             {\def\POL@result{-1.\empty{0/1[0]}}}%
             {\POL@@mul}}%
}%
\def\POL@@mul{%
  \expandafter\POL@split\POL@A;\POL@degA\POL@polA
  \expandafter\POL@split\POL@B;\POL@degB\POL@polB
  \ifnum\POL@degA>\POL@degB\relax
    \xintAssignArray\POL@polA\to\POL@arrayA
    \xintAssignArray\POL@polB\to\POL@arrayB
  \else
    \xintAssignArray\POL@polB\to\POL@arrayA
    \xintAssignArray\POL@polA\to\POL@arrayB
    \let\POL@tmp\POL@degB
    \let\POL@degB\POL@degA
    \let\POL@degA\POL@tmp
  \fi
  \count@\z@
  \xintloop
    \POL@@mul@phaseIloopbody
  \unless\ifnum\POL@degB<\count@
  \repeat
  \xintloop
  \unless\ifnum\POL@degA<\count@ % car attention au cas de mêmes degrés
    \POL@@mul@phaseIIloopbody
  \repeat
  \edef\POL@degC{\the\numexpr\POL@degA+\POL@degB}%
  \xintloop
  \unless\ifnum\POL@degC<\count@
    \POL@@mul@phaseIIIloopbody
  \repeat
  %\count@\the\numexpr\POL@degC+\@ne\relax % never zero polynomial here
  \POL@resultfromarray C%
}%
\def\POL@@mul@phaseIloopbody{%
    \advance\count@\@ne
    \def\POL@tmp{0[0]}%
    \count\tw@\z@
    \xintloop
       \advance\count\tw@\@ne
       \edef\POL@tmp{%
         \xintScalarAdd
           {\POL@tmp}%
           {\xintScalarMul
             {\@nameuse{POL@arrayA\the\count\tw@}}%
             {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}%
           }%
          }%
    \ifnum\count\tw@<\count@
    \repeat
    \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp
}%
\def\POL@@mul@phaseIIloopbody{%
    \advance\count@\@ne
    \def\POL@tmp{0[0]}%
    \count\tw@\count@
    \advance\count\tw@-\@nameuse{POL@arrayB0} %
    \xintloop
    \ifnum\count\tw@<\count@
      \advance\count\tw@\@ne
      \edef\POL@tmp{%
        \xintScalarAdd
          {\POL@tmp}%
          {\xintScalarMul
            {\@nameuse{POL@arrayA\the\count\tw@}}%
            {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}%
          }%
         }%
    \repeat
    \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp
}%
\def\POL@@mul@phaseIIIloopbody{%
    \advance\count@\@ne
    \def\POL@tmp{0[0]}%
    \count\tw@\count@
    \advance\count\tw@-\@nameuse{POL@arrayB0} %
    \xintloop
      \advance\count\tw@\@ne
      \edef\POL@tmp{%
        \xintScalarAdd{\POL@tmp}%
          {\xintScalarMul
            {\@nameuse{POL@arrayA\the\count\tw@}}%
            {\@nameuse{POL@arrayB\the\numexpr\count@+\@ne-\count\tw@}}%
          }%
         }%
    \ifnum\@nameuse{POL@arrayA0}>\count\tw@
    \repeat
    \expandafter\let\csname POL@arrayC\the\count@\endcsname\POL@tmp
}%

%% POWERS (SCALAR EXPONENT...)
\def\POL@pow #1#2{%
    \global\POL@polfalse
    \begingroup
        \def\POL@result{#1}%
        #1%
        \expandafter
    \endgroup
    \expandafter\def\expandafter\POL@A\expandafter{\POL@result}%
    \unless\ifPOL@pol
        \edef\POL@A{\xintScalarPow{#1}{#2}}% no error check
        \xintiiifZero{\POL@A}%
            {\def\POL@result{-1.\empty{0/1[0]}}}%
            {\edef\POL@result{0.\noexpand\empty{\POL@A}}}%
    \else
      \edef\POL@B{\numexpr\xintNum{#2}\relax}% no check on exponent >= 0
      \ifcase\POL@B
        \def\POL@result{0.\empty{1/1[0]}}%
      \or
        \let\POL@result\POL@A
      \else
        \POL@@pow@check
      \fi
    \fi
    \global\POL@poltrue
}%
\def\POL@@pow@check {%
% no problem here with leftover tokens!
% should I have used that I-don't-care technique more elsewhere?
    \ifnum\@ne>\POL@A
    % polynomial is a constant, must get rid of dot and \empty
        \edef\POL@A{\expandafter\xintScalarPow\romannumeral`^^@%
                    \expandafter\xint_gob_til_dot\POL@A{\POL@B}}%
        \xintiiifZero{\POL@A}%
            {\def\POL@result{-1.\empty{0/1[0]}}}%
            {\edef\POL@result{0.\noexpand\empty{\POL@A}}}%
    \else
    \ifnum\@ne=\POL@A
    % perhaps a constant times X, check constant term
        \xintiiifZero
        {\expandafter\xint_firstoftwo\romannumeral`^^@%
         \expandafter\xint_gob_til_dot\POL@A}
        {\edef\POL@result
          {\the\POL@B.% here at least 2.
           \noexpand\empty
           \romannumeral\xintreplicate{\POL@B}{{0/1[0]}}%
           {\xintScalarPow
             {\expandafter\xint_secondoftwo\romannumeral`^^@%
              \expandafter\xint_gob_til_dot\POL@A}%
             {\POL@B}}}}%
        {\POL@@pow}% not constant times X, use general recursion
    \else
        \POL@@pow% general recursion
    \fi\fi
}%
\def\POL@@pow@recurse#1#2{%
    \begingroup
        #1%
        \expandafter
    \endgroup
    \expandafter\def\expandafter\POL@A\expandafter{\POL@result}%
    \edef\POL@B{\numexpr\xintNum{#2}\relax}%
    \ifcase\POL@B
        \POL@thisshouldneverhappen
    \or
        \let\POL@result\POL@A
    \else
        \expandafter\POL@@pow
    \fi
}%
\def\POL@@pow {%
    \let\POL@pow@exp\POL@B
    \let\POL@B\POL@A
    \POL@@mul
    \let\POL@sqA\POL@result
    \ifodd\POL@pow@exp\space
       \expandafter\POL@@pow@odd
          \the\numexpr(\POL@pow@exp+\@ne)/\tw@-\@ne\expandafter.%
    \else
       \expandafter\POL@@pow@even
          \the\numexpr(\POL@pow@exp+\@ne)/\tw@-\@ne\expandafter.%
    \fi
}%
\def\POL@@pow@even#1.{%
    \expandafter\POL@@pow@recurse\expandafter
       {\expandafter\def\expandafter\POL@result\expandafter{\POL@sqA}}%
       {#1}%
}%
\def\POL@@pow@odd#1.{%
    \expandafter\POL@@pow@odd@i\expandafter{\POL@A}{#1}%
}%
\def\POL@@pow@odd@i #1#2{%
    \expandafter\POL@@pow@recurse\expandafter
       {\expandafter\def\expandafter\POL@result\expandafter{\POL@sqA}}%
       {#2}%
    \expandafter\POL@mul\expandafter
        {\expandafter\def\expandafter\POL@result\expandafter
         {\POL@result}\global\POL@poltrue}%
        {\def\POL@result{#1}\global\POL@poltrue}%
}%

%% DIVISION
%% no check on divisor being non-zero
\def\POL@div  {\POL@get\POL@A\POL@div@b}%
\def\POL@div@b{\POL@get\POL@B\POL@div@c}%
\def\POL@div@c{%
    \global\POL@poltrue
    \expandafter\POL@split\POL@A;\POL@degA\POL@polA
    \expandafter\POL@split\POL@B;\POL@degB\POL@polB
    \ifnum\POL@degA<\POL@degB\space
       \@namedef{POL@arrayQ1}{0/1[0]}%
       \def\POL@degQ{-1}%
    \else
       \xintAssignArray\POL@polA\to\POL@arrayR
       \xintAssignArray\POL@polB\to\POL@arrayB
       \POL@@div
    \fi
    \count@\numexpr\POL@degQ+\@ne\relax
    \POL@resultfromarray Q%
}%
\def\POL@@div{%
    \xintAssignArray\POL@polA\to\POL@arrayR
    \xintAssignArray\POL@polB\to\POL@arrayB
    \edef\POL@B@leading{\csname POL@arrayB\the\numexpr\POL@degB+\@ne\endcsname}%
    \edef\POL@degQ{\the\numexpr\POL@degA-\POL@degB}%
    \count@\numexpr\POL@degA+\@ne\relax
    \count\tw@\numexpr\POL@degQ+\@ne\relax
    \xintloop
      \POL@@div@loopbody
    \ifnum\count\tw@>\z@
    \repeat
    %%\expandafter\def\csname POL@arrayR0\endcsname{1}%
    \xintloop
    \xintiiifZero{\csname POL@arrayR\the\count@\endcsname}%
      {\iftrue}%
      {\iffalse}%
    \advance\count@\m@ne
    \repeat
    \edef\POL@degR{\the\numexpr\count@-\@ne}%
}%
\def\POL@@div@loopbody{%
    \edef\POL@@div@ratio{%
          \xintScalarDiv{\csname POL@arrayR\the\count@\endcsname}%
                        {\POL@B@leading}}%
    \expandafter\let\csname POL@arrayQ\the\count\tw@\endcsname
          \POL@@div@ratio
    \advance\count@\m@ne
    \advance\count\tw@\m@ne
    \count4 \count@
    \count6 \POL@degB\space
    \xintloop
    \ifnum\count6>\z@
      \expandafter\edef\csname POL@arrayR\the\count4\endcsname
        {\xintScalarSub
            {\csname POL@arrayR\the\count4\endcsname}%
            {\xintScalarMul
              {\POL@@div@ratio}%
              {\csname POL@arrayB\the\count6\endcsname}}}%
      \advance\count4 \m@ne
      \advance\count6 \m@ne
    \repeat
}%

%% MINUS SIGN AS UNARY OPERATOR
\def\POL@opp #1{%
    \global\POL@polfalse
    \begingroup
        \def\POL@result{#1}%
        #1%
        \expandafter
    \endgroup
    \expandafter\def\expandafter\POL@A\expandafter{\POL@result}%
    \unless\ifPOL@pol
        \edef\POL@A{\xintScalarOpp{#1}}%
        \xintiiifZero{\POL@A}%
            {\def\POL@result{-1.\empty{0/1[0]}}}%
            {\edef\POL@result{0.\noexpand\empty{\POL@A}}}%
    \else
      \edef\POL@B{0.\noexpand\empty{-1/1[0]}}%
      \POL@@mul
    \fi
    \global\POL@poltrue
}%


%% EXPANDABLE MACROS
\def\POL@eval@fork#1\At#2#3\krof{#2}%
\newcommand\PolEval[3]{\romannumeral`^^@\POL@eval@fork
     #2\PolEvalAt
     \At\PolEvalAtExpr\krof {#1}{#3}%
}%
\newcommand\PolEvalAt[2]
    {\xintpraw{\csname XINT_expr_userfunc_#1\endcsname{#2}}}%
\newcommand\POL@eval[2]
    {\csname XINT_expr_userfunc_#1\endcsname{#2}}%
\newcommand\PolEvalAtExpr[2]{\xinttheexpr #1(#2)\relax}%
%
\newcommand\PolEvalReduced[3]{\romannumeral`^^@\POL@eval@fork
     #2\PolEvalReducedAt
     \At\PolEvalReducedAtExpr\krof {#1}{#3}%
}%
\newcommand\PolEvalReducedAt[2]{%
    \xintpraw % in order not to print denominator if the latter equals 1
    {\xintIrr{\csname XINT_expr_userfunc_#1\endcsname{#2}}[0]}%
}%
\newcommand\PolEvalReducedAtExpr[2]{%
    \xintpraw
    {\xintIrr{\romannumeral`^^@\xintthebareeval#1(#2)\relax}[0]}%
}%
%
\newcommand\PolFloatEval[3]{\romannumeral`^^@\POL@eval@fork
     #2\PolFloatEvalAt
     \At\PolFloatEvalAtExpr\krof {#1}{#3}%
}%
\newcommand\PolFloatEvalAt[2]
    {\xintpfloat{\csname XINT_flexpr_userfunc_#1\endcsname{#2}}}%
\newcommand\PolFloatEvalAtExpr[2]{\xintthefloatexpr #1(#2)\relax}%


\newcommand\PolSturmIntervalIndex[3]{\the\numexpr\POL@eval@fork
     #2\PolSturmIntervalIndexAt
     \At\PolSturmIntervalIndexAtExpr\krof {#1}{#3}%
}%
\newcommand\PolSturmIntervalIndexAtExpr[2]
    {\PolSturmIntervalIndexAt{#1}{\xinttheexpr#2\relax}}%
\newcommand\PolSturmIntervalIndexAt[2]
    {\expandafter\POL@sturm@index@at\romannumeral`^^@#2!{#1}\xint_bye\relax}%
\def\POL@sturm@index@at#1!#2%
{%
    \expandafter\POL@sturm@index@at@iloop 
    \romannumeral`^^@\PolSturmNbOfIsolatedZeros{#2}!{#2}{#1}%
}%
% implementation is sub-optimal as it should use some kind of binary tree
% search rather than comparing to the intervals from right to left as here
\def\POL@sturm@index@at@iloop #1!%
{%
    \ifnum #1=\z@ 0\expandafter\xint_bye\fi
    \POL@sturm@index@at@iloop@a #1!%
}%
\def\POL@sturm@index@at@iloop@a #1!#2#3%
{% #1 = index, #2 = sturmname, #3 value
    \PolSturmIfZeroExactlyKnown{#2}{#1}
    {\xintifCmp{#3}{\POL@xintexprGetVar{#2L_#1}}%
       {}%
       {#1\xint_bye}%
       {0\xint_bye}%
    }%
    {\xintifGt{#3}{\POL@xintexprGetVar{#2L_#1}}%
       {\xintifLt{#3}{\POL@xintexprGetVar{#2R_#1}}%
        {#1\xint_bye}%
        {0\xint_bye}%
       }%
       {}%
    }%
    % catcode of ! is 11 in polexpr.sty
    \expandafter\POL@sturm@index@at@iloop\the\numexpr#1-\@ne !{#2}{#3}%
}%


\def\POL@leq@fork#1\LessThanOrEqualTo#2#3\krof{#2}%
\newcommand\PolSturmNbOfRootsOf[3]{\romannumeral`^^@\POL@leq@fork
     #2\PolNbOfRootsLessThanOrEqualTo
     \LessThanOrEqualTo\PolNbOfRootsLessThanOrEqualToExpr\krof {#1}{#3}%
}%
\newcommand\PolNbOfRootsLessThanOrEqualToExpr[2]
    {\PolNbOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}%
\newcommand\PolNbOfRootsLessThanOrEqualTo[1]{%
    \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@
      \expandafter\xint_firstofthree\expandafter0%
    \else
      \expandafter\PolNbOfRootsLessThanOrEqualTo@%
    \fi {#1}%
}%
\def\PolNbOfRootsLessThanOrEqualTo@ #1#2%
{%
    \expandafter\POL@nbofrootsleq@prep\romannumeral`^^@#2!{#1}%
}%
\def\POL@nbofrootsleq@prep#1!#2%
{%
    \expandafter\POL@nbofrootsleq@iloop\expandafter 1\expandafter !%
    \romannumeral0\xintsgn{\POL@eval{#2_0}{#1}}!%
    #1!{#2}%
}%
\def\POL@nbofrootsleq@iloop#1!#2!#3!#4%
{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname
    \xintifCmp{#3}{\POL@xintexprGetVar{#4L_#1}}%
      {\POL@nbofrootsleq@return #1-\@ne !}%
      {\POL@nbofrootsleq@return
       \PolSturmIfZeroExactlyKnown{#4}{#1}{#1}{#1-\@ne}!%
      }%
      % in third branch we are sure that if root is exactly known
      % the test \xintifLt will be negative
      {\xintifLt{#3}{\POL@xintexprGetVar{#4R_#1}}%
         {\POL@nbofrootsleq@return
            #1\ifnum#2=\xintSgn{\POL@eval{#4_0}{\POL@xintexprGetVar{#4L_#1}}}
              -\@ne\fi !%
         }%
         {\ifnum#1=\PolSturmNbOfIsolatedZeros{#4}
            \expandafter\POL@nbofrootsleq@rightmost
          \fi \expandafter\POL@nbofrootsleq@iloop \the\numexpr\@ne+%
         }%
       }%
    #1!#2!#3!{#4}%
}%
\def\POL@nbofrootsleq@return #1!#2!#3!#4!#5{\the\numexpr #1\relax}%
\def\POL@nbofrootsleq@rightmost\expandafter\POL@nbofrootsleq@iloop 
    \the\numexpr\@ne+#1!#2!#3!#4{#1}%


\newcommand\PolSturmNbWithMultOfRootsOf[3]
{\the\numexpr0\POL@leq@fork
     #2\PolNbWithMultOfRootsLessThanOrEqualTo
     \LessThanOrEqualTo\PolNbWithMultOfRootsLessThanOrEqualToExpr\krof {#1}{#3}%
}%
\newcommand\PolNbWithMultOfRootsLessThanOrEqualToExpr[2]
    {\PolNbWithMultOfRootsLessThanOrEqualTo{#1}{\xinttheexpr#2\relax}}%
\newcommand\PolNbWithMultOfRootsLessThanOrEqualTo[1]{%
    \ifnum\PolSturmNbOfIsolatedZeros{#1}=\z@
      \expandafter\POL@nbwmofroots@noroots
    \else
      \expandafter\PolNbWithMultOfRootsLessThanOrEqualTo@%
    \fi {#1}%
}%
\def\POL@nbwmofroots@noroots#1#2{\relax}%
\def\PolNbWithMultOfRootsLessThanOrEqualTo@ #1#2%
{%
    \expandafter\POL@nbwmofrootsleq@prep\romannumeral`^^@#2!{#1}%
}%
\def\POL@nbwmofrootsleq@prep#1!#2%
{%
    \expandafter\POL@nbwmofrootsleq@iloop\expandafter 1\expandafter !%
    \romannumeral0\xintsgn{\POL@eval{#2_0}{#1}}!%
    #1!{#2}%
}%
\def\POL@nbwmofrootsleq@iloop#1!#2!#3!#4%
{% #1 = index, #2 = sign of evaluation at value, #3 = value, #4 = sturmname
    \xintifCmp{#3}{\POL@xintexprGetVar{#4L_#1}}%
      {\POL@nbwmofrootsleq@return !}%
      {\POL@nbwmofrootsleq@return
       \PolSturmIfZeroExactlyKnown{#4}{#1}%
        {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}}{}!%
      }%
      % in third branch we are sure that if root is exactly known
      % the test \xintifLt will be negative
      {\xintifLt{#3}{\POL@xintexprGetVar{#4R_#1}}%
         {\POL@nbwmofrootsleq@return
            \unless
            \ifnum#2=\xintSgn{\POL@eval{#4_0}{\POL@xintexprGetVar{#4L_#1}}}
            +\PolSturmIsolatedZeroMultiplicity{#4}{#1}\fi !%
         }%
         {+\PolSturmIsolatedZeroMultiplicity{#4}{#1}%
          \ifnum#1=\PolSturmNbOfIsolatedZeros{#4}
             \expandafter\POL@nbwmofrootsleq@return\expandafter !%
          \fi
          \expandafter\POL@nbwmofrootsleq@iloop \the\numexpr\@ne+%
         }%
       }%
    #1!#2!#3!{#4}%
}%
\def\POL@nbwmofrootsleq@return #1!#2!#3!#4!#5{#1\relax}%


\newcommand\PolLeadingCoeff[1]{%
    \romannumeral`^^@\expandafter\expandafter\expandafter\xintlastitem
                     \expandafter\expandafter\expandafter
                     {\csname POLuserpol@#1\endcsname}%
}%
%
\newcommand\PolNthCoeff[2]{\romannumeral`^^@%
    \expandafter\POL@nthcoeff
    \romannumeral0\xintnthelt{\ifnum\numexpr#2<\z@#2\else(#2)+1\fi}%
                  {\expandafter\expandafter\expandafter
                        \xint_gob_til_dot\csname POLuserpol@#1\endcsname}@%
}%
\def\POL@nthcoeff#1@{\if @#1@\expandafter\xint_firstoftwo
                  \else\expandafter\xint_secondoftwo\fi
                  {0/1[0]}{#1}}%
%
% returns -1 for zero polynomial for context of numerical expression
% should it return -\infty?
\newcommand\PolDegree[1]{\romannumeral`^^@\expandafter\expandafter\expandafter
                         \POL@degree\csname POLuserpol@#1\endcsname;}%
\def\POL@degree #1.#2;{#1}%
%
\newcommand\PolToList[1]{\romannumeral`^^@\expandafter\expandafter\expandafter
                         \xint_gob_til_dot\csname POLuserpol@#1\endcsname}%
%
\newcommand\PolToCSV[1]{\romannumeral0\xintlistwithsep{, }{\PolToList{#1}}}%


\newcommand\PolToExprCmd[1]{\xintPRaw{\xintRawWithZeros{#1}}}%
\newcommand\PolToFloatExprCmd[1]{\xintFloat{#1}}%
\let\PolToExprTermPrefix\PolTypesetCmdPrefix
\newcommand\PolToExprOneTermStyleA[2]{%
    \ifnum#2=\z@
      \PolToExprCmd{#1}%
    \else
      \xintifOne{\xintiiAbs{#1}}
        {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix
        {\PolToExprCmd{#1}\PolToExprTimes}%
    \fi
    \ifcase\xintiiAbs{#2} %<-- space here mandatory
         \or\PolToExprVar
       \else\PolToExprVar^\xintiiAbs{#2}%
    \fi
}%
\let\PolToExprOneTerm\PolToExprOneTermStyleA
\newcommand\PolToExprOneTermStyleB[2]{%
    \ifnum#2=\z@
      \xintNumerator{#1}%
    \else
      \xintifOne{\xintiiAbs{\xintNumerator{#1}}}
        {\xintiiifSgn{#1}{-}{}{}}% + from \PolToExprTermPrefix
        {\xintNumerator{#1}\PolToExprTimes}%
    \fi
    \ifcase\xintiiAbs{#2} %<-- space here mandatory
         \or\PolToExprVar
       \else\PolToExprVar^\xintiiAbs{#2}%
    \fi
    \xintiiifOne{\xintDenominator{#1}}{}{/\xintDenominator{#1}}%
}%
\newcommand\PolToFloatExprOneTerm[2]{%
    \ifnum#2=\z@
      \PolToFloatExprCmd{#1}%
    \else
      \PolToFloatExprCmd{#1}\PolToExprTimes
    \fi
    \ifcase\xintiiAbs{#2} %<-- space here mandatory
         \or\PolToExprVar
       \else\PolToExprVar^\xintiiAbs{#2}%
    \fi
}%
\newcommand\PolToExprTimes{*}%
\newcommand\PolToExprVar{x}%
\newcommand\PolToExpr[1]{%
    \if*\noexpand#1\expandafter\xint_firstoftwo\else
    \expandafter\xint_secondoftwo\fi
    \PolToExprAscending\PolToExprDescending{#1}}%
\newcommand\PolToFloatExpr[1]{%
    \if*\noexpand#1\expandafter\xint_firstoftwo\else
    \expandafter\xint_secondoftwo\fi
    \PolToFloatExprAscending\PolToFloatExprDescending{#1}}%
\newcommand\PolToExprAscending[2]{%
   \expandafter\POL@toexpr\csname POLuserpol@#2\endcsname
   \PolToExprOneTerm\POL@toexprA}%
\newcommand\PolToFloatExprAscending[2]{%
   \expandafter\POL@toexpr\csname POLuserpol@#2\endcsname
   \PolToFloatExprOneTerm\POL@toexprA}%
\newcommand\PolToExprDescending[1]{%
   \expandafter\POL@toexpr\csname POLuserpol@#1\endcsname
   \PolToExprOneTerm\POL@toexprD}%
\newcommand\PolToFloatExprDescending[1]{%
   \expandafter\POL@toexpr\csname POLuserpol@#1\endcsname
   \PolToFloatExprOneTerm\POL@toexprD}%
%
\def\POL@toexpr#1#2#3{\expandafter\POL@toexpr@
                      \expandafter#3\expandafter#2#1\relax}%
\def\POL@toexpr@#1#2#3.{%
    \ifnum#3<\z@
       #2{0/1[0]}{0}\expandafter\xint_gobble_v
    \else
       \expandafter#1%
    \fi {#3}#2}%
%
\def\POL@toexprA #1#2\empty#3{%
    \ifpoltoexprall\expandafter\POL@toexprall@b
              \else\expandafter\POL@toexpr@b
    \fi {#3}#2{0}1.%
}%
\def\POL@toexprD #1#2#3\relax{% #3 has \empty to prevent brace removal
    \expandafter\POL@toexprD@a\expandafter#2%
    \the\numexpr #1\expandafter.\romannumeral0\xintrevwithbraces{#3}\relax
}%
\def\POL@toexprD@a #1#2.#3{%
    \ifpoltoexprall\expandafter\POL@toexprall@b
              \else\expandafter\POL@toexpr@b
    \fi{#3}#1{-#2}\the\numexpr\@ne+-#2.%
}%
\def\POL@toexpr@b #1#2#3{%
    \xintiiifZero{#1}%
      {\expandafter\POL@toexpr@loop\expandafter\POL@toexpr@b}%
      {#2{#1}{#3}%
      \expandafter\POL@toexpr@loop\expandafter\POL@toexpr@c}%
   \expandafter#2%
}%
\def\POL@toexpr@c #1#2#3{%
    \xintiiifZero{#1}%
      {}%
      {\PolToExprTermPrefix{#1}#2{#1}{#3}}%
    \expandafter\POL@toexpr@loop\expandafter\POL@toexpr@c
    \expandafter#2%
}%
\def\POL@toexprall@b #1#2#3{%
    #2{#1}{#3}%
    \expandafter\POL@toexpr@loop\expandafter\POL@toexprall@c
    \expandafter#2%
}%
\def\POL@toexprall@c #1#2#3{%
    \PolToExprTermPrefix{#1}#2{#1}{#3}%
    \expandafter\POL@toexpr@loop\expandafter\POL@toexprall@c
    \expandafter#2%
}%
\def\POL@toexpr@loop#1#2#3.#4{%
  \if\relax#4\expandafter\xint_gob_til_dot\fi
  #1{#4}#2{#3}\the\numexpr\@ne+#3.%
}%


\POL@restorecatcodes
\endinput