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RegularChains[FastArithmeticTools]

  

NormalizePolynomialDim0

  

normalize a polynomial w.r.t a 0-dim regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

NormalizePolynomialDim0(f, rc, R)

Parameters

R

-

a polynomial ring

rc

-

a regular chain of R

f

-

polynomial of R

Description

• 

The command NormalizePolynomialDim0 returns a normalized form of f w.r.t. rc, that is, a polynomial q which is associated to f modulo rc, such that q is normalized w.r.t. rc.

• 

rc is zero-dimensional regular chain, and f together with rc forms a zero-dimensional 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:

p962592769:

varsy,x:

RPolynomialRingvars,p:

We consider two bivariate polynomials and want to compute their common solutions

f1xy2+y+1+2:

f2x+1y2+y+1+x3+x+1:

We first compute their subresultant chain using FFT techniques

SCubeSubresultantChainSpecializationCubef1,f2,y,R,1

SCube:=subresultant_chain_specialization_cube

(1)

We deduce their resultants

r2ResultantBySpecializationCubef1,f2,x,SCube,R

r2:=x8+2x6+962592767x5+962592766x4+962592767x3+962592766x2+4x+4

(2)

We observe below that no root of r2 cancels the leading coefficients of f1 or f2. Hence, any roots of r2 can be extended into a solution of the system by a GCD computation.

Gcdr2,xx+1modp

1

(3)

We define the regular chain consisting of r2

rcChainr2,EmptyR,R

rc:=regular_chain

(4)

We compute the GCD of f1 and f2 modulo r2

g2RegularGcdBySpecializationCubef1,f2,rc,SCube,R

g2:=x3+xy2+xy+y2+2x+y+2,regular_chain,x3+xy2+xy+y2+2x+y+2,regular_chain

(5)

We normalize this GCD w.r.t. r2 which leads to a simpler expression with one as leading coefficient  

NormalizePolynomialDim0g211,g212,R

x3+y2+x+y

(6)

See Also

NormalForm

NormalFormDim0

NormalizeRegularChainDim0

ReduceCoefficientsDim0

RegularChains

 


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