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|
--- @class PolynomialRing
--- Represents an element of a polynomial ring.
--- @field coefficients table<number, Ring>
--- @field symbol SymbolExpression
--- @field ring RingIdentifier
PolynomialRing = {}
local __PolynomialRing = {}
-- Metatable for ring objects.
local __obj = {__index = PolynomialRing, __eq = function(a, b)
return a["ring"] == b["ring"] and
(a["child"] == b["child"] or a["child"] == nil or b["child"] == nil) and
(a["symbol"] == b["symbol"] or a["child"] == nil or b["child"] == nil)
end, __tostring = function(a)
if a.child and a.symbol then return tostring(a.child) .. "[" .. a.symbol .. "]" else return "(Generic Polynomial Ring)" end
end}
--------------------------
-- Static functionality --
--------------------------
--- Creates a new ring with the given symbol and child ring.
--- @param symbol SymbolExpression
--- @param child RingIdentifier
--- @return RingIdentifier
function PolynomialRing.makering(symbol, child)
local t = {ring = PolynomialRing}
t.symbol = symbol
t.child = child
t = setmetatable(t, __obj)
return t
end
-- Shorthand constructor for a polynomial ring with integer or integer mod ring coefficients.
function PolynomialRing.R(symbol, modulus)
if modulus then
return PolynomialRing.makering(symbol, IntegerModN.makering(modulus))
end
return PolynomialRing.makering(symbol, Integer.getring())
end
--- Returns the GCD of two polynomials in a ring, assuming both rings are euclidean domains.
--- @param a PolynomialRing
--- @param b PolynomialRing
--- @return PolynomialRing
function PolynomialRing.gcd(a, b)
if a.symbol ~= b.symbol then
error("Cannot take the gcd of two polynomials with different symbols")
end
while b ~= Integer.zero() do
a, b = b, a % b
end
return a // a:lc()
end
-- Returns the GCD of two polynomials in a ring, assuming both rings are euclidean domains.
-- Also returns bezouts coefficients via extended gcd.
--- @param a PolynomialRing
--- @param b PolynomialRing
--- @return PolynomialRing, PolynomialRing, PolynomialRing
function PolynomialRing.extendedgcd(a, b)
local oldr, r = a, b
local olds, s = Integer.one(), Integer.zero()
local oldt, t = Integer.zero(), Integer.one()
while r ~= Integer.zero() do
local q = oldr // r
oldr, r = r, oldr - q*r
olds, s = s, olds - q*s
oldt, t = t, oldt - q*t
end
return oldr // oldr:lc(), olds // oldr:lc(), oldt // oldr:lc()
end
-- Returns the resultant of two polynomials in the same ring, whose coefficients are all part of a field.
--- @param a PolynomialRing
--- @param b PolynomialRing
--- @return Field
function PolynomialRing.resultant(a, b)
if a.ring == PolynomialRing.getring() or b.ring == PolynomialRing.getring() then
return PolynomialRing.resultantmulti(a, b)
end
local m, n = a.degree, b.degree
if n == Integer.zero() then
return b.coefficients[0]^m
end
local r = a % b
if r == Integer.zero() then
return r.coefficients[0]
end
local s = r.degree
local l = b:lc()
return Integer(-1)^(m*n) * l^(m-s) * PolynomialRing.resultant(b, r)
end
-- Returns the resultant of two polynomials in the same ring, whose coefficients are not part of a field.
--- @param a PolynomialRing
--- @param b PolynomialRing
--- @return Ring
function PolynomialRing.resultantmulti(a, b)
local m, n = a.degree, b.degree
if m < n then
return Integer(-1) ^ (m * n) * PolynomialRing.resultantmulti(b, a)
end
if n == Integer.zero() then
return b.coefficients[0]^m
end
local delta = m - n + Integer(1)
local _ , r = PolynomialRing.pseudodivide(a, b)
if r == Integer.zero() then
return r.coefficients[0]
end
local s = r.degree
local w = Integer(-1)^(m*n) * PolynomialRing.resultant(b, r)
local l = b:lc()
local k = delta * n - m + s
local f = l ^ k
return w // f
end
-- Given two polynomials a and b, returns a list of the remainders generated by the monic Euclidean algorithm.
--- @param a PolynomialRing
--- @param b PolynomialRing
--- @return table<number, Ring>
function PolynomialRing.monicgcdremainders(a, b)
if a.symbol ~= b.symbol then
error("Cannot take the gcd of two polynomials with different symbols")
end
local remainders = {a / a:lc(), b / b:lc()}
while true do
local q = remainders[#remainders - 1] // remainders[#remainders]
local c = remainders[#remainders - 1] - q*remainders[#remainders]
if c ~= Integer.zero() then
remainders[#remainders+1] = c/c:lc()
else
break
end
end
return remainders
end
-- Returns the partial fraction decomposition of the rational function g/f
-- given g, f, and some (not nessecarily irreducible) factorization of f.
-- If the factorization is omitted, the irreducible factorization is used.
-- The degree of g must be less than the degree of f.
--- @param g PolynomialRing
--- @param f PolynomialRing
--- @param ffactors Expression
--- @return Expression
function PolynomialRing.partialfractions(g, f, ffactors)
if g.degree >= f.degree then
error("Argument Error: The degree of g must be less than the degree of f.")
end
-- Converts f to a monic polynomial.
g = g * f:lc()
f = f / f:lc()
ffactors = ffactors or f:factor()
local expansionterms = {}
for _, factor in ipairs(ffactors.expressions) do
local k
local m
if factor.getring and factor:getring() == PolynomialRing:getring() then
m = factor
k = Integer.one()
elseif not factor:isconstant() then
m = factor.expressions[1]
k = factor.expressions[2]
end
if not factor:isconstant() then
-- Uses Chinese Remainder Theorem for each factor to determine the numerator of the term in the decomposition
local mk = m^k
local v = g % mk
local _, minv, _ = PolynomialRing.extendedgcd(f // mk, mk)
local c = v*minv % mk
if k == Integer.one() then
expansionterms[#expansionterms+1] = BinaryOperation.ADDEXP({BinaryOperation.DIVEXP({c, BinaryOperation.POWEXP({m, Integer.one()})})})
else
-- Uses the p-adic expansion of c to split terms with repeated roots.
local q = c
local r
local innerterms = {}
for i = k:asnumber(), 1, -1 do
q, r = q:divremainder(m)
innerterms[#innerterms+1] = BinaryOperation.DIVEXP({r, BinaryOperation.POWEXP({m, Integer(i)})})
end
expansionterms[#expansionterms+1] = BinaryOperation.ADDEXP(innerterms)
end
end
end
return BinaryOperation.ADDEXP(expansionterms)
end
----------------------------
-- Instance functionality --
----------------------------
-- So we don't have to copy the Euclidean operations each time
__PolynomialOperations = Copy(__EuclideanOperations)
__PolynomialOperations.__index = PolynomialRing
__PolynomialOperations.__tostring = function(a)
local out = ""
local loc = a.degree:asnumber()
while loc >= 0 do
if a.ring == PolynomialRing.getring() or (a.ring == Rational.getring() and a.ring.symbol) then
out = out .. "(" .. tostring(a.coefficients[loc]) .. ")" .. a.symbol .. "^" .. tostring(math.floor(loc)) .. "+"
else
out = out .. tostring(a.coefficients[loc]) .. a.symbol .. "^" .. tostring(math.floor(loc)) .. "+"
end
loc = loc - 1
end
return string.sub(out, 1, string.len(out) - 1)
end
__PolynomialOperations.__div = function(a, b)
if not b.getring then
return BinaryOperation.DIVEXP({a, b})
end
if Ring.resultantring(a.ring, b:getring()) ~= Ring.resultantring(a:getring(), b:getring()) then
return a:div(b:inring(Ring.resultantring(a:getring(), b:getring())))
end
if b.ring and b:getring() == Rational:getring() and a.symbol == b.ring.symbol then
return a:inring(Ring.resultantring(a:getring(), b:getring())):div(b)
end
if a:getring() == b:getring() then
return Rational(a, b, true)
end
-- TODO: Fix this for arbitrary depth
if a:getring() == PolynomialRing:getring() and b:getring() == PolynomialRing:getring() and a.symbol == b.symbol then
local oring = Ring.resultantring(a:getring(), b:getring())
return Rational(a:inring(oring), b:inring(oring), true)
end
return BinaryOperation.DIVEXP({a, b})
end
function PolynomialRing:tolatex()
local out = ''
local loc = self.degree:asnumber()
if loc == 0 then
return self.coefficients[loc]:tolatex()
end
if self.ring == Rational.getring() or self.ring == Integer.getring() or self.ring == IntegerModN.getring() then
if self.coefficients[loc] ~= Integer.one() then
if self.coefficients[loc] == Integer(-1) then
out = out .. "-" .. self.symbol
elseif self.coefficients[loc] == Integer.zero() then
goto lead
else
out = out .. self.coefficients[loc]:tolatex() .. self.symbol
end
else
out = out .. self.symbol
end
if loc ~=1 then
out = out .. "^{" .. loc .. "}"
end
::lead::
loc = loc -1
while loc >=0 do
local coeff = self.coefficients[loc]
if coeff == Integer.one() then
if loc == 0 then
out = out .. "+" .. coeff:tolatex()
goto skip
else
out = out .. "+"
goto continue
end
end
if coeff == Integer(-1) then
if loc == 0 then
out = out .. "-" .. coeff:neg():tolatex()
goto skip
else
out = out .. "-"
goto continue
end
end
if coeff < Integer.zero() then
out = out .. "-" .. coeff:neg():tolatex()
end
if coeff == Integer.zero() then
goto skip
end
if coeff > Integer.zero() then
out = out .. "+" .. coeff:tolatex()
end
::continue::
if loc > 1 then
out = out .. self.symbol .. "^{" .. loc .. "}"
end
if loc == 1 then
out = out .. self.symbol
end
::skip::
loc = loc-1
end
else
while loc >=0 do
if loc >=1 then
out = out .. self.coefficients[loc]:tolatex() .. self.symbol .. "^{" .. loc .. "} + "
else
out = out .. self.coefficients[loc]:tolatex() .. self.symbol .. "^{" .. loc .. "}"
end
loc = loc-1
end
end
return out
end
function PolynomialRing:isatomic()
--if self.degree >= Integer.one() then
-- return false
--else
return false
--end
end
--test
-- Creates a new polynomial ring given an array of coefficients and a symbol
function PolynomialRing:new(coefficients, symbol, degree)
local o = {}
o = setmetatable(o, __PolynomialOperations)
if type(coefficients) ~= "table" then
error("Sent parameter of wrong type: Coefficients must be in an array")
end
o.coefficients = {}
o.degree = degree or Integer(-1)
if type(symbol) ~= "string" and not symbol.symbol then
error("Symbol must be a string")
end
o.symbol = symbol.symbol or symbol
-- Determines what ring the polynomial ring should have as its child
for index, coefficient in pairs(coefficients) do
if type(index) ~= "number" then
error("Sent parameter of wrong type: Coefficients must be in an array")
end
if not coefficient.getring then
error("Sent parameter of wrong type: Coefficients must be elements of a ring")
end
if not o.ring then
o.ring = coefficient:getring()
else
local newring = coefficient:getring()
local combinedring = Ring.resultantring(o.ring, newring)
if combinedring == newring then
o.ring = newring
elseif not o.ring == combinedring then
error("Sent parameter of wrong type: Coefficients must all be part of the same ring")
end
end
end
if not coefficients[0] then
-- Constructs the coefficients when a new polynomial is instantiated as an array
for index, coefficient in ipairs(coefficients) do
o.coefficients[index - 1] = coefficient
o.degree = o.degree + Integer.one()
end
else
-- Constructs the coefficients from an existing polynomial of coefficients
local loc = o.degree:asnumber()
while loc > 0 do
if not coefficients[loc] or coefficients[loc] == coefficients[loc]:zero() then
o.degree = o.degree - Integer.one()
else
break
end
loc = loc - 1
end
while loc >= 0 do
o.coefficients[loc] = coefficients[loc]
loc = loc - 1
end
end
-- Each value of the polynomial greater than its degree is implicitly zero
o.coefficients = setmetatable(o.coefficients, {__index = function (table, key)
return o:zeroc()
end})
return o
end
-- Returns the ring this object is an element of
function PolynomialRing:getring()
local t = {ring = PolynomialRing}
if self then
t.child = self.ring
t.symbol = self.symbol
end
t = setmetatable(t, __obj)
return t
end
-- Explicitly converts this element to an element of another ring
function PolynomialRing:inring(ring)
-- Faster equality check
if ring == self:getring() then
return self
end
if ring == Rational:getring() and ring.symbol then
return Rational(self:inring(ring.child), self:inring(ring.child):one(), true)
end
if ring.symbol == self.symbol then
local out = {}
for i = 0, self.degree:asnumber() do
out[i + 1] = self.coefficients[i]:inring(ring.child)
end
return PolynomialRing(out, self.symbol)
end
-- TODO: Allow re-ordering of polynomial rings, so from R[x][y] -> R[y][x] for instance
if ring == PolynomialRing:getring() then
return PolynomialRing({self:inring(ring.child)}, ring.symbol)
end
error("Unable to convert element to proper ring.")
end
-- Returns whether the ring is commutative
function PolynomialRing:iscommutative()
return true
end
function PolynomialRing:add(b)
local larger
if self.degree > b.degree then
larger = self
else
larger = b
end
local new = {}
local loc = 0
while loc <= larger.degree:asnumber() do
new[loc] = self.coefficients[loc] + b.coefficients[loc]
loc = loc + 1
end
return PolynomialRing(new, self.symbol, larger.degree)
end
function PolynomialRing:neg()
local new = {}
local loc = 0
while loc <= self.degree:asnumber() do
new[loc] = -self.coefficients[loc]
loc = loc + 1
end
return PolynomialRing(new, self.symbol, self.degree)
end
function PolynomialRing:mul(b)
-- Grade-school multiplication is actually faster up to a very large polynomial size due to Lua's overhead.
local new = {}
local sd = self.degree:asnumber()
local bd = b.degree:asnumber()
for i = 0, sd+bd do
new[i] = self:zeroc()
for j = math.max(0, i-bd), math.min(sd, i) do
new[i] = new[i] + self.coefficients[j]*b.coefficients[i-j]
end
end
return PolynomialRing(new, self.symbol, self.degree + b.degree)
-- return PolynomialRing(PolynomialRing.mul_rec(self.coefficients, b.coefficients), self.symbol, self.degree + b.degree)
end
-- Performs Karatsuba multiplication without constructing new polynomials recursively
function PolynomialRing.mul_rec(a, b)
if #a==0 and #b==0 then
return {[0]=a[0] * b[0], [1]=Integer.zero()}
end
local k = Integer.ceillog(Integer.max(Integer(#a), Integer(#b)) + Integer.one(), Integer(2))
local n = Integer(2) ^ k
local m = n / Integer(2)
local nn = n:asnumber()
local mn = m:asnumber()
local a0, a1, b0, b1 = {}, {}, {}, {}
for e = 0, mn - 1 do
a0[e] = a[e] or Integer.zero()
a1[e] = a[e + mn] or Integer.zero()
b0[e] = b[e] or Integer.zero()
b1[e] = b[e + mn] or Integer.zero()
end
local p1 = PolynomialRing.mul_rec(a1, b1)
local p2a = Copy(a0)
local p2b = Copy(b0)
for e = 0, mn - 1 do
p2a[e] = p2a[e] + a1[e]
p2b[e] = p2b[e] + b1[e]
end
local p2 = PolynomialRing.mul_rec(p2a, p2b)
local p3 = PolynomialRing.mul_rec(a0, b0)
local r = {}
for e = 0, mn - 1 do
p2[e] = p2[e] - p1[e] - p3[e]
r[e] = p3[e]
r[e + mn] = p2[e]
r[e + nn] = p1[e]
end
for e = mn, nn - 1 do
p2[e] = p2[e] - p1[e] - p3[e]
r[e] = r[e] + p3[e]
r[e + mn] = r[e + mn] + p2[e]
r[e + nn] = p1[e]
end
return r
end
-- Uses synthetic division.
function PolynomialRing:divremainder(b)
local n, m = self.degree:asnumber(), b.degree:asnumber()
if m > n then
return self:zero(), self
end
local o = Copy(self.coefficients)
local lc = b:lc()
for i = n, m, -1 do
o[i] = o[i] / lc
if o[i] ~= self:zeroc() then
for j = 1, m do
o[i-j] = o[i-j] - b.coefficients[m - j] * o[i]
end
end
end
local q = {}
local r = {}
for i = 0, m-1 do
r[i] = o[i]
end
r[0] = r[0] or self:zeroc()
for i = m, #o do
q[i - m] = o[i]
end
return PolynomialRing(q, self.symbol, self.degree), PolynomialRing(r, self.symbol, Integer.max(Integer.zero(), b.degree-Integer.one()))
end
-- Performs polynomial pseudodivision of this polynomial by another in the same ring,
-- and returns both the pseudoquotient and pseudoremainder.
-- In the case where both coefficients are fields, this is equivalent to division with remainder.
function PolynomialRing:pseudodivide(b)
local p = self:zero()
local s = self
local m = s.degree
local n = b.degree
local delta = Integer.max(m - n + Integer.one(), Integer.zero())
local lcb = b:lc()
local sigma = Integer.zero()
while m >= n and s ~= Integer.zero() do
local lcs = s:lc()
p = p * lcb + self:one():multiplyDegree((m-n):asnumber()) * lcs
s = s * lcb - b * self:one():multiplyDegree((m-n):asnumber()) * lcs
sigma = sigma + Integer.one()
m = s.degree
end
if delta - sigma == Integer.zero() then
return p,s
else
return lcb^(delta - sigma) * p, lcb^(delta - sigma) * s
end
end
-- Polynomial rings are never fields, but when dividing by a polynomial by a constant we may want to use / instead of //
function PolynomialRing:div(b)
return self:divremainder(b)
end
function PolynomialRing:zero()
return self.coefficients[0]:zero():inring(self:getring())
end
function PolynomialRing:zeroc()
return self.coefficients[0]:zero()
end
function PolynomialRing:one()
return self.coefficients[0]:one():inring(self:getring())
end
function PolynomialRing:onec()
return self.coefficients[0]:one()
end
function PolynomialRing:eq(b)
for i=0,math.max(self.degree:asnumber(), b.degree:asnumber()) do
if self.coefficients[i] ~= b.coefficients[i] then
return false
end
end
return true
end
-- Returns the leading coefficient of this polynomial
function PolynomialRing:lc()
return self.coefficients[self.degree:asnumber()]
end
--- @return boolean
function PolynomialRing:isconstant()
return false
end
-- This expression is free of a symbol if and only if the symbol is not the symbol used to create the ring.
function PolynomialRing:freeof(symbol)
return symbol.symbol ~= self.symbol
end
-- Replaces each expression in the map with its value.
function PolynomialRing:substitute(map)
return self:tocompoundexpression():substitute(map)
end
-- Expands a polynomial expression. Polynomials are already in expanded form, so we just need to autosimplify.
function PolynomialRing:expand()
return self:tocompoundexpression():autosimplify()
end
function PolynomialRing:autosimplify()
return self:tocompoundexpression():autosimplify()
end
-- Transforms from array format to an expression format.
function PolynomialRing:tocompoundexpression()
local terms = {}
for exponent, coefficient in pairs(self.coefficients) do
terms[exponent + 1] = BinaryOperation(BinaryOperation.MUL, {coefficient:tocompoundexpression(),
BinaryOperation(BinaryOperation.POW, {SymbolExpression(self.symbol), Integer(exponent)})})
end
return BinaryOperation(BinaryOperation.ADD, terms)
end
-- Uses Horner's rule to evaluate a polynomial at a point
function PolynomialRing:evaluateat(x)
local out = self:zeroc()
for i = self.degree:asnumber(), 1, -1 do
out = out + self.coefficients[i]
out = out * x
end
return out + self.coefficients[0]
end
-- Multiplies this polynomial by x^n
function PolynomialRing:multiplyDegree(n)
local new = {}
for e = 0, n-1 do
new[e] = self:zeroc()
end
local loc = n
while loc <= self.degree:asnumber() + n do
new[loc] = self.coefficients[loc - n]
loc = loc + 1
end
return PolynomialRing(new, self.symbol, self.degree + Integer(n))
end
-- Returns the formal derivative of this polynomial
function PolynomialRing:derivative()
if self.degree == Integer.zero() then
return PolynomialRing({self:zeroc()}, self.symbol, Integer(-1))
end
local new = {}
for e = 1, self.degree:asnumber() do
new[e - 1] = Integer(e) * self.coefficients[e]
end
return PolynomialRing(new, self.symbol, self.degree - Integer.one())
end
-- Returns the square-free factorization of a polynomial
function PolynomialRing:squarefreefactorization()
local terms
if self.ring == Rational.getring() or self.ring == Integer.getring() then
terms = self:rationalsquarefreefactorization()
elseif self.ring == IntegerModN.getring() then
if not self.ring.modulus:isprime() then
error("Cannot compute a square-free factorization of a polynomial ring contructed from a ring that is not a field.")
end
terms = self:modularsquarefreefactorization()
end
local expressions = {self:lc()}
local j = 1
for index, term in ipairs(terms) do
if term.degree ~= Integer.zero() or term.coefficients[0] ~= Integer.one() then
j = j + 1
expressions[j] = BinaryOperation.POWEXP({term, Integer(index)})
end
end
return BinaryOperation.MULEXP(expressions)
end
-- Factors a polynomial into irreducible terms
function PolynomialRing:factor()
-- Square-free factorization over an integral domain (so a polynomial ring constructed from a field)
local squarefree = self:squarefreefactorization()
local squarefreeterms = {}
local result = {squarefree.expressions[1]}
for i, expression in ipairs(squarefree.expressions) do
if i > 1 then
-- Converts square-free polynomials with rational coefficients to integer coefficients so Rational Roots / Zassenhaus can factor them
if expression.expressions[1].ring == Rational.getring() then
local factor, integerpoly = expression.expressions[1]:rationaltointeger()
result[1] = result[1] * factor ^ expression.expressions[2]
squarefreeterms[i - 1] = integerpoly
else
squarefreeterms[i - 1] = expression.expressions[1]
end
end
end
for i, expression in ipairs(squarefreeterms) do
local terms
if expression.ring == Integer.getring() then
-- Factoring over the integers first uses the rational roots test to factor out monomials (for efficiency purposes)
local remaining, factors = expression:rationalroots()
terms = factors
-- Then applies the Zassenhaus algorithm if there entire polynomial has not been factored into monomials
if remaining ~= Integer.one() then
remaining = remaining:zassenhausfactor()
for _, exp in ipairs(remaining) do
terms[#terms+1] = exp
end
end
end
if expression.ring == IntegerModN.getring() then
-- Berlekamp factorization is used for rings with integers mod a prime as coefficients
terms = expression:berlekampfactor()
end
for _, factor in ipairs(terms) do
result[#result+1] = BinaryOperation.POWEXP({factor, squarefree.expressions[i + 1].expressions[2]})
end
end
return BinaryOperation.MULEXP(result)
end
-- Uses the Rational Root test to factor out monomials of a square-free polynomial.
function PolynomialRing:rationalroots()
local remaining = self
local roots = {}
if self.coefficients[0] == Integer.zero() then
roots[1] = PolynomialRing({Integer.zero(), Integer.one()}, self.symbol)
remaining = remaining // roots[1]
end
-- This can be slower than Zassenhaus if the digits are large enough, since factoring integers is slow
-- if self.coefficients[0] > Integer(Integer.DIGITSIZE - 1) or self:lc() > Integer(Integer.DIGITSIZE - 1) then
-- return remaining, roots
-- end
while remaining ~= Integer.one() do
:: nextfactor ::
local a = remaining.coefficients[0]
local b = remaining:lc()
local afactors = a:divisors()
local bfactors = b:divisors()
for _, af in ipairs(afactors) do
for _, bf in ipairs(bfactors) do
local testroot = Rational(af, bf, true)
if remaining:evaluateat(testroot) == Integer.zero() then
roots[#roots+1] = PolynomialRing({-testroot.numerator, testroot.denominator}, self.symbol)
remaining = remaining // roots[#roots]
goto nextfactor
end
if remaining:evaluateat(-testroot) == Integer.zero() then
roots[#roots+1] = PolynomialRing({testroot.numerator, testroot.denominator}, self.symbol)
remaining = remaining // roots[#roots]
goto nextfactor
end
end
end
break
end
return remaining, roots
end
-- Returns a list of roots of the polynomial, simplified up to cubics.
function PolynomialRing:roots()
local roots = {}
local factorization = self:factor()
for i, factor in ipairs(factorization.expressions) do
if i > 1 then
local decomp = factor.expressions[1]:decompose()
for _, poly in ipairs(decomp) do
if poly.degree > Integer(3) then
table.insert(roots,RootExpression(factor.expressions[1]))
goto nextfactor
end
end
local factorroots = RootExpression(decomp[#decomp]):autosimplify()
if factorroots == true then
return true
end
if factorroots == false then
goto nextfactor
end
local replaceroots = {}
for j = #decomp - 1,1,-1 do
for _, root in ipairs(factorroots) do
local temp = RootExpression(decomp[j]):autosimplify(root)
if temp == true then
return true
end
if factorroots == false then
goto nextfactor
end
replaceroots = JoinArrays(replaceroots, temp)
end
factorroots = replaceroots
end
roots = JoinArrays(roots, factorroots)
end
end
::nextfactor::
return roots
end
-----------------
-- Inheritance --
-----------------
__PolynomialRing.__index = Ring
__PolynomialRing.__call = PolynomialRing.new
PolynomialRing = setmetatable(PolynomialRing, __PolynomialRing)
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