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diffalg

  

differential_sprem

  

return sparse pseudo remainder of a differential polynomial

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

differential_sprem (q, L, R, 'h')

differential_sprem (q, C, 'h')

Parameters

q

-

differential polynomial in R

L

-

list or a set of differential polynomials in R

C

-

characterizable differential ideal

R

-

differential polynomial ring

h

-

(optional) name

Description

• 

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

• 

The function differential_sprem is an implementation of Ritt's reduction algorithm. It is an extension of the pseudo-remainder algorithm to differential polynomials.

• 

L is assumed to form a differentially triangular set.

• 

Let A denote L or equations(C).

• 

The function differential_sprem returns a differential polynomial r such that

  

(a) hq=modr,A. 

  

(b) No proper derivative of the leaders of the elements of A appears in r.

  

(c) The degree according to a leader of any element a of A is strictly less in r than in a.

  

(d) The differential polynomial h is a power product of factors of the  initials and the separants of the elements of A.

• 

The differential_sprem(q, L, R, 'h') calling sequence returns an error message if L  contains 0. If L contains a non zero element of the ground field of R, it returns zero.

• 

The differential_sprem(q, C, 'h') calling sequence requires that q belong to the differential ring in which C is defined.

  

The function rewrite_rules shows how the equations of C are interpreted by the pseudo-reduction algorithm.

  

Then r is zero if and only if q belongs to C.

• 

The command with(diffalg,differential_sprem) allows the use of the abbreviated form of this command.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

withdiffalg:

Differential pseudo-division by a single differential polynomial:

Rdifferential_ringderivations=x,ranking=u:

pu[]1ux,x+ux

p:=u[]1ux,x+ux

(1)

qu[]21ux+1

q:=u[]21ux+1

(2)

rdifferential_spremp,q,R,'h';h

r:=u[]3u[]2u[]+1

u[]+1u[]212

(3)

Reduction according to a characterizable differential ideal:

Rdifferential_ringderivations=x,y,ranking=u:

JRosenfeld_Groebnerxux,y2+yuy+1,R

J:=characterizable,characterizable

(4)

rewrite_rulesJ1

ux,y2=yuy+1x

(5)

qux,x,x,y,y

q:=ux,x,x,y,y

(6)

rdifferential_spremq,J1,'h'

r:=2xyuyux,y+y2uyuy,y+yuy22xux,y+yuy,y+uy

(7)

h

83x4ux,y3

(8)

belongs_tohqr,J1

true

(9)

See Also

diffalg(deprecated)

diffalg(deprecated)/belongs_to

diffalg(deprecated)/differential_algebra

diffalg(deprecated)/differential_ring

diffalg(deprecated)/Rosenfeld_Groebner

DifferentialAlgebra[DifferentialPrem]

 


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