ApCoCoA-1:Weyl.WRGBS

This article is about a function from ApCoCoA-1.

Weyl.WRGBS

Convert a Groebner basis of an ideal in Weyl algebra A_n in to its reduced Groebner Basis using corresponding implementation in ApCoCoALib.

Syntax

Weyl.WRGBS(GB:LIST):LIST

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.

This function converts Groebner basis GB of and ideal I computed by ApCoCoAServer into the reduced Groebner Basis by using the corresponding implementation in ApCoCoaLib. 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 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. Therefore, if GB is a Weyl Groebner basis then output of this function will be reduced Groebner basis.

• @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.WRGBS(L);
-------------------------------
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[1]
-------------------------------
-- Done.
-------------------------------

Example

A2::=ZZ/(13)[x[1..2],y[1..2]]; -- define appropriate ring and activate it with Use command.
Use A2;
I:=Ideal(x[1]^13-1,x[1]^3+x[1]y[1]^2+y[1]-2);

-- Done.
-------------------------------
GbI:=Weyl.WGB(I,0);  --computes complete GB of the ideal I
-------------------------------
-- CoCoAServer: computing Cpu Time = 0.063
-------------------------------
-- Done.
-------------------------------
Len(GbI);

11
-------------------------------
-- Done.
-------------------------------
GbI:=Weyl.WRGBS(GbI);GbI;Len(GbI);
-------------------------------
-- CoCoAServer: computing Cpu Time = 0.031
-------------------------------
[
  y[1]^10 - 5x[1]^2y[1]^6 - 3x[1]y[1]^7 - y[1]^8 + 6x[1]^2y[1]^5 - x[1]y[1]^6 + y[1]^7 + 3x[1]^2y[1]^4 - 6x[1]y[1]^5 + 2y[1]^6 +
x[1]^2y[1]^3 - 6x[1]y[1]^4 + 4y[1]^5 + x[1]^2y[1]^2 + 2x[1]y[1]^3 + 2x[1]^2y[1] + 2x[1]y[1]^2 - 2y[1]^3 + 3x[1]^2 - 3x[1]y[1] +
4y[1]^2 - 6x[1] - 6y[1] + 5,
  x[1]^2y[1]^7 - 4x[1]y[1]^8 + y[1]^9 - 6x[1]^2y[1]^6 - x[1]y[1]^7 + 2y[1]^8 + 4x[1]^2y[1]^5 - 3y[1]^7 - x[1]^2y[1]^4 - 
2x[1]y[1]^5 - 5x[1]^2y[1]^3 - 4x[1]y[1]^4 + 4x[1]^2y[1]^2 - 3x[1]y[1]^3 - 6y[1]^4 - 5x[1]^2y[1] + 6x[1]y[1]^2 + y[1]^3 + 3x[1]^2 - 
5x[1]y[1] + 3y[1]^2 + 4x[1] + y[1] - 5,
  x[1]y[1]^9 + 6x[1]y[1]^8 + 2y[1]^9 + 6x[1]^2y[1]^6 + 4x[1]y[1]^7 - 6y[1]^8 - 3x[1]^2y[1]^5 + 4x[1]y[1]^6 - 3x[1]^2y[1]^4 - 
4x[1]y[1]^5 + 6y[1]^6 - 6x[1]y[1]^4 + 3y[1]^5 + x[1]^2y[1]^2 - 6x[1]y[1]^3 + y[1]^4 - 2x[1]^2y[1] + 6x[1]y[1]^2 - y[1]^3 + 
3x[1]^2 - 2x[1]y[1] + 6y[1]^2 - 5x[1] + 2y[1] + 5,
  x[1]^3 + x[1]y[1]^2 + y[1] - 2]
-------------------------------
4  -- which is now size of reduced GB of the ideal I
-------------------------------
-- Done.
-------------------------------

See also

Weyl.WNormalForm

Weyl.WGB

Weyl.WRGB

Weyl.WRedGB

Introduction to Groebner Basis in CoCoA

Introduction to CoCoAServer