ApCoCoA-1:Weyl.WRGB

This article is about a function from ApCoCoA-1.

Weyl.WRGB

Reduced Groebner basis of an ideal I in Weyl algebra A_n.

Syntax

Weyl.WRGB(GB:LIST):LIST

Description

This function converts a Weyl Groebner basis GB computed by ApCoCoAServer into the reduced Weyl Groebner Basis. If GB is not a Groebner basis then the output will not be the reduced Groebner basis. In fact, this function reduces a list GB of Weyl polynomials using Weyl.WNR into a new list L such that Ideal(L) = Ideal(GB), every polynomial is reduced with respect to the remaining polynomials in the list L and leading coefficient of each polynomial in L is 1.

• @param GB Groebner Basis of an ideal in the Weyl algebra.

• @result The reduced Groebner Basis of the given ideal.

Example

A1::=QQ[x,d];	--Define appropriate ring
Use A1;
L:=[x,d,1];
Weyl.WRGB(L);
[1]
-------------------------------

Example

A2::=ZZ/7[x[1..2],y[1..2]]; -- define appropriate ring
Use A2;
I:=Ideal(2x[1]^14y[1]^7,x[1]^2y[1]^3+x[1]^2-1,y[2]^7-1,x[2]^3y[2]^2-x[2]y[2]-3x[2]-1);
GbI:=Weyl.WGB(I,0);Len(GbI);

-------------------------------
-- CoCoAServer: computing Cpu Time = 0.485
-------------------------------
42   -- size of complete GB of the ideal I
-------------------------------
Time GbI:=Weyl.WRGB(GbI);Len(GbI);

Cpu time = 9.61, User time = 10
-------------------------------
11
-------------------------------
-- Done.
-------------------------------
Time GbI:=Weyl.WRGBS(GbI);Len(GbI); -- Weyl.WRGBS() can now be used for calling same implementation in ApCoCoALib
                                    -- note that this speeds up the computations
-------------------------------
-- CoCoAServer: computing Cpu Time = 0  
-------------------------------
Cpu time = 0.04, User time = 0   
-------------------------------
11   -- this is now size of reduced GB of the ideal I
-------------------------------
-- Done.
-------------------------------

See also

Weyl.WNormalForm

Weyl.WGB

Weyl.WRGBS

Weyl.WRedGB

Introduction to Groebner Basis in CoCoA

Introduction to CoCoAServer