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torusInvariants -- ring of invariants

Synopsis

Description

Let T=(K*)r be the r-dimensional torus acting on the polynomial ring R=K[X1,...,Xn] diagonally. Such an action can be described as follows: there are integers aij, i=1,...,r, j=1,...,n, such that (λ1,...,λr)∈T acts by the substitution

Xj→λ1a1j*...*λrarjXj, j=1,...,n.

In order to compute the ring of invariants RT, one must specify the matrix (aij).
i1 : R=QQ[x,y,z,w];
i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}});

              2        4
o2 : Matrix ZZ  <--- ZZ
i3 : torusInvariants(T,R)

             2           2
o3 = ideal (x z, x*y*z, y z)

o3 : Ideal of R

Caveat

It is of course possible that RT=K. At present, Normaliz cannot deal with the zero cone and will issue the (wrong) error message that the cone is not pointed.

See also

Ways to use torusInvariants :