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D.4.16.6 torusInvariants
Procedure from library normaliz.lib (see normaliz_lib).
- Usage:
- torusInvariants(intmat A);
- Return:
- Returns an ideal representing the list of monomials generating the ring of
invariants as an algebra over the coefficient field.
@tex
$R^T$.
@end tex
The function returns the ideal given by the input matrix A if one of
the options supp , triang , or hvect has been
activated.
However, in this case some numerical invariants are computed, and
some other data may be contained in files that you can read into
Singular.
- Background:
- @tex
Let $T = (K^*)^r$ be the $r$-dimensional torus acting on the polynomial ring
$R = K[X_1 ,\ldots,X_n]$ diagonally. Such an action can be described as
follows: there are integers $a_{i,j}$, $i=1,\ldots,r$, $j=1,\ldots,n$, such
that $(\lambda_1,\ldots,\lambda_r)\in T$ acts by the substitution
$$ X_j \mapsto \lambda_1^{a_{1,j}} \cdots \lambda_r^{a_{r,j}}X_j,
\quad j=1,\ldots,n.$$
In order to compute the ring of invariants $R^T$ one must specify the matrix
$A=(a_{i,j})$.
@end tex
Example:
| LIB "normaliz.lib";
ring R=0,(x,y,z,w),dp;
intmat E[2][4] = -1,-1,2,0, 1,1,-2,-1;
torusInvariants(E);
==> _[1]=x2z
==> _[2]=xyz
==> _[3]=y2z
| See also:
diagInvariants;
finiteDiagInvariants;
intersectionValRingIdeals;
intersectionValRings.
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