ApCoCoA-1:Weyl.WNR
| This article is about a function from ApCoCoA-1. |
Weyl.WNR
Computes the normal remainder of a Weyl polynomial F with respect
to a polynomial or a list of Weyl polynomials using corresponding implementation in ApCoCoALib.
Syntax
Weyl.WNR(F:POLY,G:POLY):POLY Weyl.WNR(F:POLY,G:LIST):POLY
Description
Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.
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. That is, a Weyl polynomial P belongs to an ideal I iff Weyl.WNR(P,Weyl.WGB(I))=0.
@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.WNR(F1,L); ------------------------------- -- CoCoAServer: computing Cpu Time = 0 ------------------------------- 0 ------------------------------- -- Done. ------------------------------- Weyl.WNR(F1,Gens(Ideal(F2,F3))); ------------------------------- -- CoCoAServer: computing Cpu Time = 0 ------------------------------- -d[1]^3d[2]^5d[3]^5 + x[2]^5 ------------------------------- -- Done. ------------------------------- Weyl.WNR(x[2]^5-d[1]^3,L); ------------------------------- -- CoCoAServer: computing Cpu Time = 0 ------------------------------- x[2]^5 - d[1]^3 ------------------------------- -- Done. ------------------------------- Weyl.WNR(x[2]^5-d[1]^3d[2]^7d[3]^6,F1); ------------------------------- -- CoCoAServer: computing Cpu Time = 0 ------------------------------- -x[2]^5d[2]^2d[3] - 3x[2]^4d[2]d[3] + x[2]^5 + x[2]^3d[3] ------------------------------- -- Done. -------------------------------
Example
Use A1::=QQ[x,d]; Weyl.WNR(xd,d); -- CoCoAServer: computing Cpu Time = 0 ------------------------------- 0 ------------------------------- -- Done. ------------------------------- Weyl.WNR(xd,x); -- CoCoAServer: computing Cpu Time = 0 ------------------------------- -1 ------------------------------- -- Done. -------------------------------
See also
