iterated resultant of a polynomial w.r.t a 0-dim regular chain - Maple Help

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RegularChains[FastArithmeticTools][IteratedResultantDim0] - iterated resultant of a polynomial w.r.t a 0-dim regular chain

Calling Sequence

IteratedResultantDim0(f, rc, R)

Parameters

R

-

a polynomial ring

rc

-

a regular chain

f

-

a polynomial

Description

• 

The function call IteratedResultantDim0(f, rc, R) returns the iterated resultant of f w.r.t. rc. See the command IteratedResultant for a definition of the notion of an iterated resultant.

• 

rc is assumed to be a zero-dimensional normalized regular chain.

• 

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.

Examples

withRegularChains:

withFastArithmeticTools:

withChainTools:

Define a ring of polynomials.

p:=962592769;vars:=x1,x2,x3,x4:R:=PolynomialRingvars,p:

p:=962592769

(1)

Randomly generating (dense) regular chain and polynomial

N:=nopsvars:dg:=3:degs:=seq4,i=1..N:pol:=randpolyvars,dense,degree=dg+randmodpmodp;tc:=RandomRegularChainDim0vars,degs,p;Equationstc,R

pol:=962592762x13+22x12x2+962592714x12x3+962592675x12x4+962592713x1x22+962592707x1x2x4+962592696x1x32+962592765x1x3x4+962592759x1x42+80x23+962592725x22x3+71x22x4+962592694x2x32+962592759x2x3x4+962592729x2x42+23x33+75x32x4+6x3x42+37x43+87x12+97x1x2+962592686x1x3+62x1x4+962592752x22+962592762x2x3+42x2x4+962592677x32+74x3x4+962592746x42+962592687x1+962592719x2+72x3+87x4+874547123

tc:=regular_chain

x14+962592759x13+962592687x2+71x3+16x4+83x12+9x22+962592709x3+962592686x4+98x2+962592721x32+962592750x4+62x3+37x42+5x4+96x1+962592752x23+25x3+91x4x22+98x32+962592705x4+64x3+962592679x42+962592709x4+962592735x2+962592756x33+44x4+962592767x32+71x42+962592722x4+962592730x3+962592716x43+962592697x42+962592672x4+91831581,x24+x23+x3+55x4+962592741x22+16x32+30x4+962592742x3+962592754x42+962592710x4+962592673x2+72x33+962592682x4+47x32+962592679x42+43x4+92x3+962592678x43+962592681x42+962592721x4+614095058,x34+11x33+962592720x4+962592722x32+40x42+962592688x4+91x3+68x43+962592759x42+31x4+175602554,x44+962592746x43+10x42+962592708x4+685457535

(2)

Compute the iterated resultant of pol w.r.t. tc

r1:=IteratedResultantDim0pol,tc,R

r1:=446889812

(3)

Compare with the generic algorithm (non-fast and non-modular algorithm) of the command IteratedResultant.

r2:=IteratedResultantpol,tc,R

r2:=446889812

(4)

Check that the two results match.

Expandr1r2modp

0

(5)

See Also

IteratedResultant, IteratedResultantDim1, RandomRegularChainDim0, RandomRegularChainDim1, RegularChains, ResultantBySpecializationCube, SubresultantChainSpecializationCube


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