# ApCoCoA-1:Weyl.WNormalRemainder

From ApCoCoAWiki

## 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