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Weyl.TwoWGB

Computes the reduced two-sided Groebner basis of a two-sided ideal I in the Weyl algebra A_n over the field of positive characteristic.
Syntax
          
Weyl.TwoWGB(I:IDEAL, L: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 computes the reduced two-sided Groebner Basis for a two-sided Ideal I = (f_1,f_2, ..., f_r) where every generator f_i should be a Weyl polynomial in standard form. The function first computes a left Groebner basis of the ideal I and then tries to update it into the two-sided one. One can also use another parameter L, a list of distinct positive integers corresponding to the number of indeterminate (1,2,..., 2n) to be eliminated while computing the Groebner basis of the ideal I. The following parameters are optional:

Example
A1::=QQ[x,d];	--Define an appropriate ring
Use A1;
I:=Ideal(x,d);  -- Now start ApCoCoA server for executing next command
Weyl.TwoWGB(I);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[1]
-------------------------------


Example
A1::=ZZ/(3)[x,d];	--Define an appropriate ring
Use A1;
I1:=Ideal(x,d);  -- Now start ApCoCoA server for executing next command
Weyl.TwoWGB(I1);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[1]
-------------------------------
I2:=Ideal(x^3);
Weyl.TwoWGB(I2);
[x^3]
-------------------------------
-- Done.
-------------------------------
I3:=Ideal(x^3d-d^3+1,d^6+d^3+1);
Weyl.TwoWGB(I3);
[d^3 - 1, x^3]
-------------------------------
-- Done.
-------------------------------


Example
W3::=ZZ/(7)[x[1..3],d[1..3]];
Use W3;
I2:=Ideal(x[1]^7x[2],d[1]^7-x[1]^7-3);
Weyl.TwoWGB(I2);

[x[1]^7, d[1]^7 - 3]
-------------------------------
-- Done.
-------------------------------
I3:=Ideal(x[1]^7x[2]^7-x[3]^14-1,d[1]^7-x[1]^7,x[3]^7d[3]^7+3);
Set Indentation;
Weyl.TwoWGB(I3);

[
  x[2]^7d[1]^7d[3]^7 + 3x[3]^7 - d[3]^7,
  x[3]^14 - x[2]^7d[1]^7 + 1,
  x[3]^7d[3]^7 + 3,
  x[1]^7 - d[1]^7]
-------------------------------
-- Done.
-------------------------------


See Also