NormalFormDim0 - Maple Help
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RegularChains[FastArithmeticTools]

  

NormalFormDim0

  

normal form of a polynomial w.r.t. a 0-dim regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

NormalFormDim0(f, rc, R)

Parameters

R

-

polynomial ring

rc

-

a regular chain of R

f

-

polynomial of R

Description

• 

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 2e divides p1.  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.

Examples

withRegularChains:

withChainTools:

withFastArithmeticTools:

p962592769:

varsx,y,z:

Nnopsvars:

RPolynomialRingvars,p:

c1z2+286748183z+134705551:

c2y2+914706789yz+686506773y+823875308z+417988453:

c3x2+224047618xyz+197329999xy+838274835xz+792563861x+600038529yz+434770098y+251400283z+918968185:

lfc1,c2,c3:

tcChainlf,EmptyR,R:

lfx+c1c2c3,c1c2c1c3,1+c1c2c3:

dg10:

q1randpolyvars,dense,degree=dg+randmodpmodp:

q2randpolyvars,dense,degree=dg+randmodpmodp:

q3randpolyvars,dense,degree=dg+randmodpmodp:

rrandpolyvars,dense,degree=dg+randmodpmodp:

fq1c1+q2c2+q3c3+r

f610303894+z2+286748183z+134705551962592762x10+22x9y+962592714x9z+87x8y2+962592713x8yz+962592707x8z2+962592765x7y3+962592686x7y2z+62x7yz2+962592725x7z3+962592759x6y4+962592762x6y3z+42x6y2z2+75x6yz3+72x6z4+29x5y5+98x5y4z+10x5y3z2+962592740x5y2z3+962592722x5yz4+962592759x5z5+x4y6+55x4y5z+16x4y4z2+962592754x4y3z3+962592682x4y2z4+962592678x4yz5+13x4z6+962592709x3y7+962592686x3y6z+962592721x3y5z2+37x3y4z3+25x3y3z4+64x3y2z5+962592767x3yz6+33x3z7+50x2y8+962592709x2y7z+7x2y6z2+34x2y5z3+52x2y4z4+69x2y3z5+962592734x2y2z6+962592700x2yz7+962592759x2z8+962592747xy9+51xy8z+50xy7z2+962592722xy6z3+25xy5z4+10xy4z5+31xy3z6+962592718xy2z7+962592733xyz8+81xz9+962592766y10+962592713y9z+962592699y8z2+962592748y7z3+962592718y6z4+45y5z5+86y4z6+62y3z7+61y2z8+962592698yz9+962592709z10+962592675x9+97x8z+962592759x7y2+962592687x7yz+71x7z2+962592729x6y3+962592719x6y2z+962592677x6yz2+37x6z3+962592746x5y4+962592708x5y3z+95x5y2z2+40x5yz3+31x5z4+962592741x4y5+30x4y4z+962592710x4y3z2+47x4y2z3+962592681x4yz4+962592759x4z5+98x3y6+962592750x3y5z+5x3y4z2+91x3y3z3+962592679x3y2z4+71x3yz5+10x3z6+962592727x2y7+962592680x2y6z+962592701x2y5z2+962592749x2y4z3+80x2y3z4+97x2y2z5+962592723x2yz6+962592704x2z7+962592742xy8+60xy7z+962592672xy6z2+31xy5z3+962592763xy4z4+73xy3z5+88xy2z6+962592712xyz7+65xz8+962592678y9+42y8z+962592742y7z2+16y6z3+49y5z4+962592672y4z5+96y3z6+962592767y2z7+962592703yz8+7z9+962592696x8+80x7y+962592752x7z+23x6y2+6x6yz+962592746x6z2+962592761x5y3+11x5y2z+962592688x5yz2+962592718x5z3+962592742x4y4+962592673x4y3z+962592679x4y2z2+962592721x4yz3+962592687x4z4+62x3y5+96x3y4z+962592709x3y2z3+962592722x3yz4+7x3z5+962592699x2y6+962592709x2y5z+962592765x2y4z2+28x2y3z3+30x2y2z4+962592736x2yz5+962592684x2z6+962592678xy7+962592767xy6z+962592742xy5z2+80xy4z3+95xy3z4+97xy2z5+85xyz6+962592757xz7+9y8+962592690y7z+962592684y6z2+962592711y5z3+962592755y4z4+962592718y3z5+86y2z6+962592735yz7+962592708z8+962592694x7+74x6y+87x6z+962592720x5y2+91x5yz+77x5z2+72x4y3+43x4y2z+53x4yz2+71x4z3+962592752x3y4+98x3y3z+962592735x3y2z2+962592730x3yz3+962592680x3z4+16x2y5+962592680x2y4z+962592727x2y3z2+962592705x2y2z3+87x2yz4+54x2z5+962592738xy6+65xy5z+962592685xy4z2+68xy3z3+962592702xy2z4+80xyz5+78xz6+962592747y7+962592725y6z+49y5z2+83y4z3+89y3z4+57y2z5+72yz6+45z7+44x6+68x5y+95x5z+92x4y2+962592741x4yz+16x4z2+962592705x3y3+962592756x3y2z+962592716x3yz2+65x3z3+962592692x2y4+962592736x2y3z+89x2y2z2+962592735x2yz3+88xy5+57xy4z+962592740xy3z2+58xy2z3+90xyz4+5xz5+962592738y6+y5z+962592673y4z2+14y3z3+962592734y2z4+962592729yz5+962592719z6+x5+5x4y+83x4z+44x3y2+962592697x3yz+12x3z2+21x2y3+962592753x2y2z+40x2yz2+18x2z3+962592720xy4+5xy3z+29xy2z2+74xyz3+962592706xz4+962592674y5+962592761y4z+962592690y3z2+57y2z3+962592701yz4+962592747z5+9x4+962592672x3y+962592744x3z+59x2y2+77x2yz+52x2z2+962592743xy3+37xy2z+27xyz2+962592764xz3+962592715y4+962592711y3z+28y2z2+962592754yz3+48z4+962592673x3+x2y+36x2z+5xy2+9xyz+36xz2+962592674y3+63y2z+962592737yz2+962592687z3+91x2+962592678xy+962592761xz+21y2+17yz+46z2+30x+87y+962592720z+700802342+y2+914706789yz+686506773y+823875308z+417988453962592747x10+38x9y+962592721x9z+962592682x8y2+962592756x8yz+962592706x8z2+66x7y3+88x7y2z+962592730x7yz2+962592717x7z3+962592747x6y4+17x6y3z+21x6y2z2+962592695x6yz3+962592704x6z4+26x5y5+13x5y4z+962592750x5y3z2+48x5y2z3+6x5yz4+83x5z5+45x4y6+962592683x4y5z+962592748x4y4z2+35x4y3z3+31x4y2z4+962592710x4yz5+962592702x4z6+962592759x3y7+11x3y6z+962592768x3y5z2+88x3y4z3+962592733x3y3z4+52x3y2z5+94x3yz6+60x3z7+67x2y8+962592697x2y7z+4x2y6z2+962592686x2y5z3+962592723x2y4z4+962592705x2y3z5+962592757x2y2z6+962592724x2yz7+962592689x2z8+5xy9+4xy8z+962592715xy7z2+962592731xy6z3+80xy5z4+962592678xy4z5+33xy3z6+962592687xy2z7+962592689xyz8+962592710xz9+34y10+962592765y9z+962592688y8z2+42y7z3+962592742y6z4+81y5z5+962592759y4z6+92y3z7+2y2z8+962592749yz9+89z10+962592682x8y+962592738x8z+74x7y2+962592749x7yz+37x7z2+63x6y3+962592757x6y2z+55x6yz2+962592741x6z3+83x5y4+962592707x5y3z+962592685x5y2z2+76x5yz3+962592686x5z4+962592681x4y5+28x4y4z+962592738x4y3z2+962592717x4y2z3+962592722x4yz4+962592707x4z5+962592713x3y6+49x3y5z+962592706x3y4z2+962592693x3y3z3+x3y2z4+6x3yz5+962592679x3z6+2x2y7+962592685x2y6z+63x2y4z3+8x2y3z4+962592767x2y2z5+962592735x2yz6+42x2z7+962592680xy8+962592688xy7z+962592719xy6z2+14xy5z3+962592721xy4z4+962592682xy3z5+50xy2z6+962592673xyz7+83xz8+31y9+98y8z+962592732y7z2+11y6z3+39y5z4+61y4z5+962592767y3z6+7y2z7+962592714yz8+30z9+90x8+31x7y+962592710x7z+962592748x6y2+962592752x6yz+962592764x6z2+962592757x5y3+962592686x5y2z+95x5yz2+53x5z3+92x4y4+962592752x4y3z+44x4y2z2+58x4yz3+70x4z4+962592742x3y5+61x3y4z+962592747x3y3z2+87x3y2z3+21x3yz4+11x3z5+962592709x2y6+84x2y5z+27x2y4z2+43x2y3z3+962592724x2y2z4+94x2yz5+86x2z6+22xy7+962592694xy6z+962592713xy5z2+53xy4z3+24xy3z4+88xy2z5+962592697xyz6+962592766xz7+962592754y8+67y7z+72y6z2+76y5z3+962592712y4z4+64y3z5+24y2z6+962592705yz7+962592727z8+962592715x7+83x6y+27x6z+96x5y2+962592755x5yz+962592763x5z2+17x4y3+962592753x4y2z+962592713x4yz2+93x4z3+46x3y4+962592714x3y3z+11x3y2z2+18x3yz3+9625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