return sparse pseudo remainder of a differential polynomial
differential_sprem (q, L, R, 'h')
differential_sprem (q, C, 'h')
differential polynomial in R
list or a set of differential polynomials in R
characterizable differential ideal
differential polynomial ring
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
(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.
Differential pseudo-division by a single differential polynomial:
R ≔ differential_ring⁡derivations=x,ranking=u:
p ≔ u−1⁢ux,x+ux
q ≔ u2−1⁢ux+1
r ≔ differential_sprem⁡p,q,R,'h';h
r ≔ −u3−u2−u+1
Reduction according to a characterizable differential ideal:
R ≔ differential_ring⁡derivations=x,y,ranking=u:
J ≔ Rosenfeld_Groebner⁡x⁢ux,y2+y⁢uy+1,R
J ≔ characterizable,characterizable
q ≔ ux,x,x,y,y
r ≔ differential_sprem⁡q,J1,'h'
r ≔ −2⁢x⁢y⁢uy⁢ux,y+y2⁢uy⁢uy,y+y⁢uy2−2⁢x⁢ux,y+y⁢uy,y+uy
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