ApCoCoA-1:Weyl.AnnFs

Weyl.AnnFs

Computes annihilating ideal of a polynomial F^s in Weyl algebra A_n.

Syntax

Weyl.AnnFs(F:POLY):IDEAL

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 annihilating ideal of a polynomial F^s using the Algorithm of Oaku and Takayama, where F is a polynomial in Weyl algebra D. F should not involve any of the indeterminates in {y1, ..., yn}. This ideal belongs to the Weyl algebra A_s =D[s]= QQ[x1, ..., xn, y1, ..., yn, s,w] where s commutes with all x_i and y_i's and w is redundant indeterminate used just to create internal structure of the weyl algebra.

• @param F A polynomial F in the indeterminates x1, ..., xn of a Weyl Algebra D.

• @return An ideal in A_s=QQ[x1, ..., xn,y1, ...,yn, s,w].

Example

A2::=QQ[x[1..2],d[1..2]]; --Define appropriate ring
Use A2;
F:=x[1]^3-x[2]^2;
-------------------------------
AnnI:=Weyl.AnnFs(F);
-- CoCoAServer: computing Cpu Time = 0.078
-------------------------------
Ideal of Ring A_s = QQ[x[1..2],y[1..2],s,w]
Where current indeterminates are mapped into ring A_s as follows:
x[1] --> x[1] and d[1] --> y[1]
x[2] --> x[2] and d[2] --> y[2]

-------------------------------
AnnI;
A_s :: Ideal(
  3x[1]^2y[2] + 2x[2]y[1],
  2x[1]y[1] + 3x[2]y[2] - 6s)   --AnnI belongs to the new ring A_s
-------------------------------

Example

A3::=QQ[x[1..3],d[1..3]]; --Define appropriate ring
Use A3;

F:=x[2]^2-x[1]x[3]-1;
Weyl.AnnFs(F);
-- CoCoAServer: computing Cpu Time = 0.14
-------------------------------
Ideal of Ring A_s = QQ[x[1..3],y[1..3],s,w]
A_s :: Ideal(2x[2]y[1] + x[3]y[2], x[1]y[1] - x[3]y[3], x[2]x[3]y[2] + 2x[3]^2y[3] - 2x[3]s + 2y[1], x[2]^2y[2] + 2x[2]x[3]y[3] - 2x[2]s - y[2], -x[2]^2y[3] + x[1]x[3]y[3] - x[1]s + y[3], x[1]y[2] + 2x[2]y[3])
-------------------------------

See also

Introduction to CoCoAServer