The command ReduceCoefficientsDim0 returns the normal form of f w.r.t. rc in the sense of Groebner bases.

rc is assumed to be a normalized zero-dimensional regular chain and all variables of f but the main one must be algebraic w.r.t. rc. See the subpackage ChainTools for more information about these concepts.

R must have a prime characteristic $p$ such that FFT-based polynomial arithmetic can be used for this computation. The higher the degrees of f and rc are, the larger must be $e$ such that $2^e$ divides $p-1$.  If the degree of f or rc is too large, then an error is raised.

The algorithm relies on the fast division trick (based on power series inversion) and FFT-based multivariate multiplication.

$\mathrm{with}\left(\mathrm{RegularChains}\right)$

$\left[\mathrm{AlgebraicGeometryTools}\,\mathrm{ChainTools}\,\mathrm{ConstructibleSetTools}\,\mathrm{Display}\,\mathrm{DisplayPolynomialRing}\,\mathrm{Equations}\,\mathrm{ExtendedRegularGcd}\,\mathrm{FastArithmeticTools}\,\mathrm{Inequations}\,\mathrm{Info}\,\mathrm{Initial}\,\mathrm{Intersect}\,\mathrm{Inverse}\,\mathrm{IsRegular}\,\mathrm{LazyRealTriangularize}\,\mathrm{MainDegree}\,\mathrm{MainVariable}\,\mathrm{MatrixCombine}\,\mathrm{MatrixTools}\,\mathrm{NormalForm}\,\mathrm{ParametricSystemTools}\,\mathrm{PolynomialRing}\,\mathrm{Rank}\,\mathrm{RealTriangularize}\,\mathrm{RegularGcd}\,\mathrm{RegularizeInitial}\,\mathrm{SamplePoints}\,\mathrm{SemiAlgebraicSetTools}\,\mathrm{Separant}\,\mathrm{SparsePseudoRemainder}\,\mathrm{SuggestVariableOrder}\,\mathrm{Tail}\,\mathrm{Triangularize}\right]$

$\mathrm{with}\left(\mathrm{FastArithmeticTools}\right)$

$\left[\mathrm{BivariateModularTriangularize}\,\mathrm{IteratedResultantDim0}\,\mathrm{IteratedResultantDim1}\,\mathrm{NormalFormDim0}\,\mathrm{NormalizePolynomialDim0}\,\mathrm{NormalizeRegularChainDim0}\,\mathrm{RandomRegularChainDim0}\,\mathrm{RandomRegularChainDim1}\,\mathrm{ReduceCoefficientsDim0}\,\mathrm{RegularGcdBySpecializationCube}\,\mathrm{RegularizeDim0}\,\mathrm{ResultantBySpecializationCube}\,\mathrm{SubresultantChainSpecializationCube}\right]$

$\mathrm{with}\left(\mathrm{ChainTools}\right)$

$\left[\mathrm{Chain}\,\mathrm{ChangeOfCoordinates}\,\mathrm{ChangeOfOrder}\,\mathrm{Construct}\,\mathrm{Cut}\,\mathrm{DahanSchostTransform}\,\mathrm{Dimension}\,\mathrm{Empty}\,\mathrm{EqualSaturatedIdeals}\,\mathrm{EquiprojectableDecomposition}\,\mathrm{Extend}\,\mathrm{ExtendedNormalizedGcd}\,\mathrm{IsAlgebraic}\,\mathrm{IsEmptyChain}\,\mathrm{IsInRadical}\,\mathrm{IsInSaturate}\,\mathrm{IsIncluded}\,\mathrm{IsPrimitive}\,\mathrm{IsStronglyNormalized}\,\mathrm{IsZeroDimensional}\,\mathrm{IteratedResultant}\,\mathrm{LastSubresultant}\,\mathrm{Lift}\,\mathrm{ListConstruct}\,\mathrm{NormalizeRegularChain}\,\mathrm{NumberOfSolutions}\,\mathrm{Polynomial}\,\mathrm{Regularize}\,\mathrm{RemoveRedundantComponents}\,\mathrm{SeparateSolutions}\,\mathrm{Squarefree}\,\mathrm{SquarefreeFactorization}\,\mathrm{SubresultantChain}\,\mathrm{SubresultantOfIndex}\,\mathrm{Under}\,\mathrm{Upper}\right]$

$\mathrm{variables}≔\left[x\,y\,z\right]\;$$p≔957349889$

$\mathrm{variables}≔\left[x\,y\,z\right]$

$p≔957349889$

$\mathrm{sys}≔\left\{5y^4-3\,-20x+y-z\,-x^5+y^5-3y-1\right\}$

$\mathrm{sys}≔\left\{-20x+y-z\,5y^4-3\,-x^5+y^5-3y-1\right\}$

$R≔\mathrm{PolynomialRing}\left(\mathrm{variables}\,p\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{lrc}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\right)$

$\mathrm{lrc}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{rc}≔\mathrm{lrc}\left[1\right]$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Equations}\left(\mathrm{rc}\,R\right)$

$\left[\left(z^{12}+94127136z^8+691135635z^7+676458799z^4+195425386z^3+326553470z^2+574327669\right)x+27352854z^{13}+673373922z^9+410681381z^8+817312291z^5+308837227z^4+32655347z^3+116876413z+880926729\,\left(z^{12}+94127136z^8+691135635z^7+676458799z^4+195425386z^3+326553470z^2+574327669\right)y+547057079z^{13}+927802747z^9+821042762z^8+352188797z^5+237219820z^4+326553470z^3+805850702z+386236578\,z^{20}+957349886z^{16}+944549889z^{15}+886639826z^{12}+458149889z^{11}+156173647z^{10}+568152312z^8+120112423z^7+434195336z^6+398220483z^5+536874419z^4+604689895z^3+446611758z^2+237311560z+665813406\right]$

$\mathrm{px}≔\mathrm{Polynomial}\left(x\,\mathrm{rc}\,R\right)$

$\mathrm{px}≔x{z}^{12}+27352854z^{13}+94127136x{z}^8+673373922z^9+691135635x{z}^7+410681381z^8+676458799x{z}^4+817312291z^5+195425386x{z}^3+308837227z^4+326553470x{z}^2+32655347z^3+574327669x+116876413z+880926729$

$\mathrm{ipx}≔\mathrm{Initial}\left(\mathrm{px}\,R\right)$

$\mathrm{ipx}≔z^{12}+94127136z^8+691135635z^7+676458799z^4+195425386z^3+326553470z^2+574327669$

$\mathrm{linv}≔\mathrm{Inverse}\left(\mathrm{ipx}\,\mathrm{rc}\,R\right)$

$\mathrm{linv}≔\left[\left[\left[174020324z^{19}+197335754z^{18}+7625943z^{17}+198840137z^{16}+378204215z^{15}+815531348z^{14}+358244196z^{13}+680868023z^{12}+248247024z^{11}+563170682z^{10}+678017442z^9+232546371z^8+493675934z^7+717866054z^6+661798200z^5+439140691z^4+372603338z^3+113779500z^2+110488854z+493921163\,1\,\mathrm{regular\_chain}\right]\right]\,\left[\right]\right]$

$\mathrm{invipx}≔\mathrm{linv}\left[1\right]\left[1\right]\left[1\right]$

$\mathrm{invipx}≔174020324z^{19}+197335754z^{18}+7625943z^{17}+198840137z^{16}+378204215z^{15}+815531348z^{14}+358244196z^{13}+680868023z^{12}+248247024z^{11}+563170682z^{10}+678017442z^9+232546371z^8+493675934z^7+717866054z^6+661798200z^5+439140691z^4+372603338z^3+113779500z^2+110488854z+493921163$

$\mathrm{ReduceCoefficientsDim0}\left(\mathrm{invipx}\mathrm{px}\,\mathrm{rc}\,R\right)$

$703897958z^{19}+637307906z^{18}+745731651z^{17}+21899044z^{16}+658962013z^{15}+899050902z^{14}+77671904z^{13}+921629286z^{12}+870449919z^{11}+122035854z^{10}+791154398z^9+547395190z^8+624024465z^7+710904034z^6+4427709z^5+954705258z^4+221310023z^3+584706443z^2+x+332923317z+743851316$