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DifferentialAlgebra[Tools][DifferentialPrem] - the Ritt reduction algorithm

Calling Sequence

DifferentialPrem (p,regchain,opts)

DifferentialPrem (p,redset,R,opts)

Parameters

p

-

a differential polynomial

regchain

-

a regular differential chain

redset

-

a polynomial or a list or a set of differential polynomials

opts (optional)

-

a sequence of options

Description

• 

The function call DifferentialPrem (p,regchain) returns a sequence h,r such that h is a power product of initials and separants of regchain, r is a differential polynomial fully reduced (see below) with respect to regchain, and, hp=r modulo the differential ideal generated by the regular differential chain.

• 

The function call DifferentialPrem (p,redset,R,opts) returns a sequence h,r such that h is a power product of initials and separants of redset, r is a differential polynomial fully reduced with respect to each element of redset, and, hp=r modulo the differential ideal generated by redset. The elements of redset must depend on at least, one derivative and have integer coefficients. All the differential polynomials are regarded as elements of R, or, of its embedding ring, if R is an ideal.

• 

A differential polynomial p belongs to the differential ideal defined by regchain if and only if, the function call DifferentialPrem (p,regchain) returns a sequence h,0 whose second component is zero.

• 

This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form DifferentialPrem(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][DifferentialPrem](...).

Examples

withDifferentialAlgebra:withTools:

R:=DifferentialRingderivations=x,y,blocks=v,u

R:=differential_ring

(1)

syst:=ux24u,ux,yvyu+1,vx,xux

syst:=ux24u,ux,yvyu+1,vx,xux

(2)

ideal:=RosenfeldGroebnersyst,R1

ideal:=regular_differential_chain

(3)

p:=ux,y

p:=ux,y

(4)

h,r:=DifferentialPremp,ideal

h,r:=2uy,2ux

(5)

h,r:=DifferentialPremp,syst,R

h,r:=2ux,4uy

(6)

The ratio r/h is equivalent to p modulo the differential ideal defined by the regular differential chain, but, it is not the normal form of p 

NormalFormrh,ideal,NormalFormp,ideal

12uxuyu,12uxuyu

(7)

Different modes of reduction are available

h,r:=DifferentialPremux3,syst,R,reduction=partial

h,r:=1,ux3

(8)

h,r:=DifferentialPremux,y+ux3,syst,reduction=algebraic,R

h,r:=vy,4uuxvy+u1

(9)

See Also

DifferentialAlgebra, LeadingDerivative, LeadingRank, NormalForm, BelongsTo


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