ApCoCoA-1:Weyl.CharI

Weyl.CharI

Computes the characteristic ideal of a D-ideal I in Weyl algebra A_n.

Syntax

Weyl.CharI(I:IDEAL):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 the characteristic ideal, InIw(I,[0,e]), of a D-ideal I the Weyl algebra D. This is an ideal in the commutative polynomial ring in 2n variables. Note that this information will also be displayed along with the output. The zero set of this ideal in affine 2n-space is called characteristic variety of I. Due to limitations in CoCoA4, [0,e] is replaced by [1,1000]. This will be modified later in the future release with CoCoA5.

• @param I An ideal in the Weyl algebra.

• @return Characteristic ideal of the given ideal.

Example

A2::=QQ[x[1..2],d[1..2]]; --Define appropriate ring
Use A2;
F:=x[1]^3-x[2]^2;
I:=Ideal(F*d[1]+Der(F,x[1]),F*d[2]+Der(F,x[2]));
ChI:=Weyl.CharI(I);ChI;
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
The characteristic ideal lies in QQ[x[1..2],d[1..2]]
--------------------------------------------------

-------------------------------
Ideal(x[1]^3d[2] - x[2]^2d[2], x[1]^3d[1] - x[2]^2d[1])
-------------------------------
Weyl.IsHolonomic(I);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
FALSE     -- the D-ideal I is not holonomic having dim=3
-------------------------------
Weyl.WDim(I);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
3
-------------------------------
Weyl.WDim(ChI);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
2  -- this would be wrong since ChI is not a D-ideal use Dim(A2/ChI) instead.
-------------------------------
Dim(A2/ChI);
3
-------------------------------

Example

A2::=QQ[x[1..2],d[1..2]]; --Define appropriate ring
Use A2;
I:=Ideal(x[1]d[2],x[2]d[1]);
ChI:=Weyl.CharI(I); ChI;
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
The characteristic ideal lies in QQ[x[1..2],d[1..2]]
--------------------------------------------------

-------------------------------
Ideal(x[1]d[1] - x[2]d[2], x[2]d[1], x[1]d[2])
-------------------------------
Weyl.IsHolonomic(I);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
TRUE
-------------------------------
Dim(A2/ChI);
2
-------------------------------

See also

Introduction to CoCoAServer

Weyl.InIw