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Convert a Groebner basis of an ideal in Weyl algebra
A_n in to its reduced Groebner Basis using corresponding implementation in ApCoCoALib.
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.
A1::=QQ[x,d]; --Define appropriate ring
Use A1;
L:=[x,d,1];
Weyl.WRGBS(L);
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-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[1]
-------------------------------
-- Done.
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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.
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GbI:=Weyl.WGB(I,0); --computes complete GB of the ideal I
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-- CoCoAServer: computing Cpu Time = 0.063
-------------------------------
-- Done.
-------------------------------
Len(GbI);
11
-------------------------------
-- Done.
-------------------------------
GbI:=Weyl.WRGBS(GbI);GbI;Len(GbI);
-------------------------------
-- CoCoAServer: computing Cpu Time = 0.031
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[
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
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-- Done.
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