# 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 when 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("All weights shoud be strictly positive") ------------------------------- 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]) -------------------------------

### See also