ApCoCoA-1:Weyl.WNormalRemainder

Weyl.WNormalRemainder

Computes the normal remainder of a Weyl polynomial F with respect

to a polynomial or a set of polynomials.

Syntax

Weyl.WNormalRemainder(F:POLY,G:POLY):POLY
Weyl.WNormalRemainder(F:POLY,G:LIST):POLY

Description

Computes the normal remainder of a Weyl polynomial F with respect to a polynomial G or a set of polynomials in the list G.

If G is Groebner basis then this function is used for ideal membership problem.

• @param F A Weyl polynomial in normal form.

• @param G A Weyl polynomial or a list of Weyl polynomials.

• @return The remainder as a Weyl polynomial using normal remainder algorithm in Weyl algebra A_n.

Note: All polynomials that are not in normal form should be first converted into normal form using Weyl.WNormalForm, otherwise you may get unexpected results.

Example

W3::=ZZ/(7)[x[1..3],d[1..3]];
Use W3;
F1:=-d[1]^3d[2]^5d[3]^5+x[2]^5;
F2:=-3x[2]d[2]^5d[3]^5+x[2]d[1]^3;
F3:=-2d[1]^4d[2]^5-x[1]d[2]^7+x[3]^3d[3]^5;
L:=[F1,F2,F3];
Weyl.WNormalRemainder(F1,L);
0
-------------------------------
Weyl.WNormalRemainder(F1,Gens(Ideal(F2,F3)));
-d[1]^3d[2]^5d[3]^5 + x[2]^5
-------------------------------
Weyl.WNormalRemainder(x[2]^5-d[1]^3,L);
x[2]^5 - d[1]^3
-------------------------------
Weyl.WNormalRemainder(x[2]^5-d[1]^3d[2]^7d[3]^6,F1);
-x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3]
-------------------------------

See also

Weyl.WNormalForm

Weyl.WNR