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RegularChains[FastArithmeticTools]

  

IteratedResultantDim1

  

iterated resultant of a polynomial w.r.t a one-dim regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IteratedResultantDim1(f, rc, R, v)

IteratedResultantDim1(f, rc, R, v, bound)

Parameters

R

-

a polynomial ring

rc

-

a regular chain

f

-

a polynomial

v

-

variable of R

bound

-

an upper bound of the degree of the iterated resultant to be computed (optional)

Description

• 

The function call IteratedResultantDim1(f, rc, R) returns the numerator of the iterated resultant of f w.r.t. rc, computed over the field of univariate rational functions in v and with coefficients in R. See the command IteratedResultant for a definition of the notion of an iterated resultant.

• 

rc is assumed to be a one-dimensional normalized regular chain with v as free variable and f has positive degree w.r.t. v.

• 

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 default value of bound is the product of the total degrees of the polynomials in rc and f.

• 

The iterated resultant computed by the command IteratedResultant produces the same answer provided that all initials in the regular chain rc are equal to 1.

• 

The interest of the function call IteratedResultantDim1(f, rc, R) resides in the fact that, if the polynomial f is regular modulo the saturated ideal of the regular chain rc, then the roots of the returned polynomial form the projection on the v-axis of the intersection of the hypersurface defined by f and the quasi-component defined by rc.

Examples

withRegularChains:

withFastArithmeticTools:

withChainTools:

Define a ring of polynomials.

p469762049;varsx1,x2,x3,x4;RPolynomialRingvars,p

p:=469762049

vars:=x1,x2,x3,x4

R:=polynomial_ring

(1)

Define random dense polynomial and regular chain of R.

Nnopsvars:dg3:degsseq2,i=1..N:polrandpolyvars,dense,degree=dg+randmodpmodp;tcRandomRegularChainDim1vars,degs,p;Equationstc,R

pol:=469762042x13+22x12x2+469761994x12x3+469761955x12x4+469761993x1x22+469761987x1x2x4+469761976x1x32+469762045x1x3x4+469762039x1x42+80x23+469762005x22x3+71x22x4+469761974x2x32+469762039x2x3x4+469762009x2x42+23x33+75x32x4+6x3x42+37x43+87x12+97x1x2+469761966x1x3+62x1x4+469762032x22+469762042x2x3+42x2x4+469761957x32+74x3x4+469762026x42+469761967x1+469761999x2+72x3+87x4+23102807

tc:=regular_chain

x12+469761998x1+77x2+95x3+x4+377175716,x22+40x2+469761968x3+91x4+2502552,x32+469762020x3+95x4+63792240

(2)

Compute the (numerator) of the iterated resultant

r1IteratedResultantDim1pol,tc,R,x4

r1:=68613548x424+347134095x423+360682950x422+449975966x421+452755530x420+347383754x419+223883343x418+428024257x417+190189697x416+166005727x415+88755441x414+16726876x413+30728041x412+191794x411+55677935x410+232265645x49+131365622x48+100732316x47+465359200x46+463678220x45+280061786x44+453663429x43+383524352x42+254364287x4+418973534

(3)

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

r2IteratedResultantpol,tc,R

r2:=68613548x424+347134095x423+360682950x422+449975966x421+452755530x420+347383754x419+223883343x418+428024257x417+190189697x416+166005727x415+88755441x414+16726876x413+30728041x412+191794x411+55677935x410+232265645x49+131365622x48+100732316x47+465359200x46+463678220x45+280061786x44+453663429x43+383524352x42+254364287x4+418973534

(4)

Check that the two results are equal, since here all initials are equal to 1.

Expandr1r2modp

0

(5)

See Also

IteratedResultant

IteratedResultantDim0

RandomRegularChainDim1

RegularChains

ResultantBySpecializationCube

SubresultantChainSpecializationCube

 


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