Difference between revisions of "ApCoCoA-1:Weyl.InIw"

Weyl.InIw

Computes the initial ideal of a D-ideal I in Weyl algebra A_n with respect to the weight vector W=(u_i,v_i).

Syntax

```Weyl.InIw(I:IDEAL,W:LIST):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.

Computes the initial ideal of a D-ideal I in the Weyl algebra D with respect to weight vector W:=[u,v] such that u+v > 0. Here u=(u1,...,un) and v=(v1,...,vn) are weights of indeterminates [x1,...,xn] and [y1,...,yn] respectively. Note that InIw(P,W) is an ideal of graded ring of D with respect to weight vector W. Due to limitations in CoCoA4, all u_i and v_i should be positive integers. Computation of initial ideal if u+v= 0 is not implemented yet.

• @param I An ideal in the Weyl algebra.

• @param W A list of n positive integers, where n = number of indeterminates.

• @return An ideal, which is the initial ideal of I with respect to W.

Beta Warning: This method, package or class is a beta version. It may not work as intended or its interface may change in the next version! So please be careful when you're intending to use it.

Example

```Use A4::=QQ[x[1..4],d[1..4]];
I4:=Ideal(d[2]d[3]-d[1]d[4],x[1]d[1]-x[4]d[4]-1,x[2]d[2]+x[4]d[4]+1,x[3]d[3]+x[4]d[4]+2);
Weyl.InIw(I4,[0,0,0,0,1,1,1,1]);  --This function is implemented only for positive weights!
ERROR: All weights shoud be strictly positive
CONTEXT: Error(<quotes>All weights shoud be strictly positive</quotes>)
-------------------------------
Weyl.InIw(I4,[1,1,1,1,2,2,1,1]);
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
Ideal(x[2]x[3]x[4]d[4]^2 - x[1]x[4]^2d[4]^2, x[1]x[4]d[3]d[4] + x[2]x[4]d[4]^2, x[1]x[4]d[2]d[4], x[2]d[1]d[4],
x[3]d[1]d[4] + x[4]d[2]d[4], x[1]d[1], x[2]d[2], x[3]d[3] + x[4]d[4], d[2]d[3] - d[1]d[4])
-------------------------------
Weyl.InIw(I4,[1,2,1,1,2,2,1,1]);
-- CoCoAServer: computing Cpu Time = 0.032
-------------------------------
Ideal(x[2]x[4]d[4]^2, x[1]x[4]d[2]d[4], x[2]d[1]d[4], x[3]d[1]d[4] + x[4]d[2]d[4], x[1]d[1], x[2]d[2],
x[3]d[3] + x[4]d[4], d[2]d[3] - d[1]d[4])
-------------------------------

Weyl.InIw(I4,[2,2,2,2,1,1,1,1]);
-- CoCoAServer: computing Cpu Time = 0.031
-------------------------------
Ideal(x[2]x[3]x[4]d[4]^2 - x[1]x[4]^2d[4]^2, x[1]x[4]d[3]d[4] + x[2]x[4]d[4]^2, x[1]x[4]d[2]d[4] + x[3]x[4]d[4]^2,
x[2]d[1]d[4] + x[4]d[3]d[4], x[3]d[1]d[4] + x[4]d[2]d[4], x[1]d[1] - x[4]d[4], x[2]d[2] + x[4]d[4],
x[3]d[3] + x[4]d[4], d[2]d[3] - d[1]d[4])
-------------------------------

```