Returns the  normal form of f w.r.t. rc in the sense of Groebner bases

rc must be a normalized zero-dimensional regular chain and all variables in f must be algebraic w.r.t. rc. See the subpackage ChainTools for these notions.

Moreover R must have a prime characteristic $p$ such that FFT-based polynomial arithmetic can be used for this actual 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. When both commands NormalFormDim0 and NormalForm apply, the former one will often outperform the latter one.

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

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

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

$p≔962592769\:$

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

$N≔\mathrm{nops}\left(\mathrm{vars}\right)\:$

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

$\mathrm{c1}≔z^2+286748183z+134705551\:$

$\mathrm{c2}≔y^2+914706789yz+686506773y+823875308z+417988453\:$

$\mathrm{c3}≔x^2+224047618xyz+197329999xy+838274835xz+792563861x+600038529yz+434770098y+251400283z+918968185\:$

$\mathrm{lf}≔\left[\mathrm{c1}\,\mathrm{c2}\,\mathrm{c3}\right]\:$

$\mathrm{tc}≔\mathrm{Chain}\left(\mathrm{lf}\,\mathrm{Empty}\left(R\right)\,R\right)\:$

$\mathrm{lf}≔\left[x+\mathrm{c1}\mathrm{c2}\mathrm{c3}\,\left(\mathrm{c1}-\mathrm{c2}\right)\left(\mathrm{c1}-\mathrm{c3}\right)\,1+\mathrm{c1}\left(\mathrm{c2}-\mathrm{c3}\right)\right]\:$

$\mathrm{dg}≔10\:$

$\mathrm{q1}≔\mathrm{randpoly}\left(\mathrm{vars}\,\mathrm{dense}\,\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\mathbf{mod}p\right)\mathbf{mod}p\:$

$\mathrm{q2}≔\mathrm{randpoly}\left(\mathrm{vars}\,\mathrm{dense}\,\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\mathbf{mod}p\right)\mathbf{mod}p\:$

$\mathrm{q3}≔\mathrm{randpoly}\left(\mathrm{vars}\,\mathrm{dense}\,\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\mathbf{mod}p\right)\mathbf{mod}p\:$

$r≔\mathrm{randpoly}\left(\mathrm{vars}\,\mathrm{dense}\,\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\mathbf{mod}p\right)\mathbf{mod}p\:$

$f≔\mathrm{q1}\mathrm{c1}+\mathrm{q2}\mathrm{c2}+\mathrm{q3}\mathrm{c3}+r$

$f≔610303894+\left(z^2+286748183z+134705551\right)\left(962592762x^{10}+22x^{9}y+962592714x^{9}z+87x^8{y}^2+962592713x^{8}yz+962592707x^8{z}^2+962592765x^7{y}^3+962592686x^7{y}^{2}z+62x^{7}y{z}^2+962592725x^7{z}^3+962592759x^6{y}^4+962592762x^6{y}^{3}z+42x^6{y}^2{z}^2+75x^{6}y{z}^3+72x^6{z}^4+29x^5{y}^5+98x^5{y}^{4}z+10x^5{y}^3{z}^2+962592740x^5{y}^2{z}^3+962592722x^{5}y{z}^4+962592759x^5{z}^5+x^4{y}^6+55x^4{y}^{5}z+16x^4{y}^4{z}^2+962592754x^4{y}^3{z}^3+962592682x^4{y}^2{z}^4+962592678x^{4}y{z}^5+13x^4{z}^6+962592709x^3{y}^7+962592686x^3{y}^{6}z+962592721x^3{y}^5{z}^2+37x^3{y}^4{z}^3+25x^3{y}^3{z}^4+64x^3{y}^2{z}^5+962592767x^{3}y{z}^6+33x^3{z}^7+50x^2{y}^8+962592709x^2{y}^{7}z+7x^2{y}^6{z}^2+34x^2{y}^5{z}^3+52x^2{y}^4{z}^4+69x^2{y}^3{z}^5+962592734x^2{y}^2{z}^6+962592700x^{2}y{z}^7+962592759x^2{z}^8+962592747x{y}^9+51x{y}^{8}z+50x{y}^7{z}^2+962592722x{y}^6{z}^3+25x{y}^5{z}^4+10x{y}^4{z}^5+31x{y}^3{z}^6+962592718x{y}^2{z}^7+962592733xy{z}^8+81x{z}^9+962592766y^{10}+962592713y^{9}z+962592699y^8{z}^2+962592748y^7{z}^3+962592718y^6{z}^4+45y^5{z}^5+86y^4{z}^6+62y^3{z}^7+61y^2{z}^8+962592698y{z}^9+962592709z^{10}+962592675x^9+97x^{8}z+962592759x^7{y}^2+962592687x^{7}yz+71x^7{z}^2+962592729x^6{y}^3+962592719x^6{y}^{2}z+962592677x^{6}y{z}^2+37x^6{z}^3+962592746x^5{y}^4+962592708x^5{y}^{3}z+95x^5{y}^2{z}^2+40x^{5}y{z}^3+31x^5{z}^4+962592741x^4{y}^5+30x^4{y}^{4}z+962592710x^4{y}^3{z}^2+47x^4{y}^2{z}^3+962592681x^{4}y{z}^4+962592759x^4{z}^5+98x^3{y}^6+962592750x^3{y}^{5}z+5x^3{y}^4{z}^2+91x^3{y}^3{z}^3+962592679x^3{y}^2{z}^4+71x^{3}y{z}^5+10x^3{z}^6+962592727x^2{y}^7+962592680x^2{y}^{6}z+962592701x^2{y}^5{z}^2+962592749x^2{y}^4{z}^3+80x^2{y}^3{z}^4+97x^2{y}^2{z}^5+962592723x^{2}y{z}^6+962592704x^2{z}^7+962592742x{y}^8+60x{y}^{7}z+962592672x{y}^6{z}^2+31x{y}^5{z}^3+962592763x{y}^4{z}^4+73x{y}^3{z}^5+88x{y}^2{z}^6+962592712xy{z}^7+65x{z}^8+962592678y^9+42y^{8}z+962592742y^7{z}^2+16y^6{z}^3+49y^5{z}^4+962592672y^4{z}^5+96y^3{z}^6+962592767y^2{z}^7+962592703y{z}^8+7z^9+962592696x^8+80x^{7}y+962592752x^{7}z+23x^6{y}^2+6x^{6}yz+962592746x^6{z}^2+962592761x^5{y}^3+11x^5{y}^{2}z+962592688x^{5}y{z}^2+962592718x^5{z}^3+962592742x^4{y}^4+962592673x^4{y}^{3}z+962592679x^4{y}^2{z}^2+962592721x^{4}y{z}^3+962592687x^4{z}^4+62x^3{y}^5+96x^3{y}^{4}z+962592709x^3{y}^2{z}^3+962592722x^{3}y{z}^4+7x^3{z}^5+962592699x^2{y}^6+962592709x^2{y}^{5}z+962592765x^2{y}^4{z}^2+28x^2{y}^3{z}^3+30x^2{y}^2{z}^4+962592736x^{2}y{z}^5+962592684x^2{z}^6+962592678x{y}^7+962592767x{y}^{6}z+962592742x{y}^5{z}^2+80x{y}^4{z}^3+95x{y}^3{z}^4+97x{y}^2{z}^5+85xy{z}^6+962592757x{z}^7+9y^8+962592690y^{7}z+962592684y^6{z}^2+962592711y^5{z}^3+962592755y^4{z}^4+962592718y^3{z}^5+86y^2{z}^6+962592735y{z}^7+962592708z^8+962592694x^7+74x^{6}y+87x^{6}z+962592720x^5{y}^2+91x^{5}yz+77x^5{z}^2+72x^4{y}^3+43x^4{y}^{2}z+53x^{4}y{z}^2+71x^4{z}^3+962592752x^3{y}^4+98x^3{y}^{3}z+962592735x^3{y}^2{z}^2+962592730x^{3}y{z}^3+962592680x^3{z}^4+16x^2{y}^5+962592680x^2{y}^{4}z+962592727x^2{y}^3{z}^2+962592705x^2{y}^2{z}^3+87x^{2}y{z}^4+54x^2{z}^5+962592738x{y}^6+65x{y}^{5}z+962592685x{y}^4{z}^2+68x{y}^3{z}^3+962592702x{y}^2{z}^4+80xy{z}^5+78x{z}^6+962592747y^7+962592725y^{6}z+49y^5{z}^2+83y^4{z}^3+89y^3{z}^4+57y^2{z}^5+72y{z}^6+45z^7+44x^6+68x^{5}y+95x^{5}z+92x^4{y}^2+962592741x^{4}yz+16x^4{z}^2+962592705x^3{y}^3+962592756x^3{y}^{2}z+962592716x^{3}y{z}^2+65x^3{z}^3+962592692x^2{y}^4+962592736x^2{y}^{3}z+89x^2{y}^2{z}^2+962592735x^{2}y{z}^3+88x{y}^5+57x{y}^{4}z+962592740x{y}^3{z}^2+58x{y}^2{z}^3+90xy{z}^4+5x{z}^5+962592738y^6+y^{5}z+962592673y^4{z}^2+14y^3{z}^3+962592734y^2{z}^4+962592729y{z}^5+962592719z^6+x^5+5x^{4}y+83x^{4}z+44x^3{y}^2+962592697x^{3}yz+12x^3{z}^2+21x^2{y}^3+962592753x^2{y}^{2}z+40x^{2}y{z}^2+18x^2{z}^3+962592720x{y}^4+5x{y}^{3}z+29x{y}^2{z}^2+74xy{z}^3+962592706x{z}^4+962592674y^5+962592761y^{4}z+962592690y^3{z}^2+57y^2{z}^3+962592701y{z}^4+962592747z^5+9x^4+962592672x^{3}y+962592744x^{3}z+59x^2{y}^2+77x^{2}yz+52x^2{z}^2+962592743x{y}^3+37x{y}^{2}z+27xy{z}^2+962592764x{z}^3+962592715y^4+962592711y^{3}z+28y^2{z}^2+962592754y{z}^3+48z^4+962592673x^3+x^{2}y+36x^{2}z+5x{y}^2+9xyz+36x{z}^2+962592674y^3+63y^{2}z+962592737y{z}^2+962592687z^3+91x^2+962592678xy+962592761xz+21y^2+17yz+46z^2+30x+87y+962592720z+700802342\right)+\left(y^2+914706789yz+686506773y+823875308z+417988453\right)\left(962592747x^{10}+38x^{9}y+962592721x^{9}z+962592682x^8{y}^2+962592756x^{8}yz+962592706x^8{z}^2+66x^7{y}^3+88x^7{y}^{2}z+962592730x^{7}y{z}^2+962592717x^7{z}^3+962592747x^6{y}^4+17x^6{y}^{3}z+21x^6{y}^2{z}^2+962592695x^{6}y{z}^3+962592704x^6{z}^4+26x^5{y}^5+13x^5{y}^{4}z+962592750x^5{y}^3{z}^2+48x^5{y}^2{z}^3+6x^{5}y{z}^4+83x^5{z}^5+45x^4{y}^6+962592683x^4{y}^{5}z+962592748x^4{y}^4{z}^2+35x^4{y}^3{z}^3+31x^4{y}^2{z}^4+962592710x^{4}y{z}^5+962592702x^4{z}^6+962592759x^3{y}^7+11x^3{y}^{6}z+962592768x^3{y}^5{z}^2+88x^3{y}^4{z}^3+962592733x^3{y}^3{z}^4+52x^3{y}^2{z}^5+94x^{3}y{z}^6+60x^3{z}^7+67x^2{y}^8+962592697x^2{y}^{7}z+4x^2{y}^6{z}^2+962592686x^2{y}^5{z}^3+962592723x^2{y}^4{z}^4+962592705x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