Saturations of Subalgebras, SAGBI Bases, and U-invariants

TL;DR

This paper develops algorithms for saturating subalgebras with respect to an element, especially in graded cases, and applies these methods to compute invariants in polynomial rings, extending classical invariant theory.

Contribution

It introduces a procedure to compute saturations of subalgebras, proves the commutation of SAGBI basis computation and saturation under certain conditions, and applies these results to invariant theory.

Findings

Algorithm for saturating subalgebras with respect to an element.

Proof that SAGBI basis computation and saturation commute under specific conditions.

Application of techniques to compute U-invariants in polynomial rings.

Abstract

Given a polynomial ring $P$ over a field $K$, an element $g \in P$, and a $K$-subalgebra $S$ of $P$, we deal with the problem of saturating $S$ with respect to $g$, i.e. computing $Sat_g(S) = S[g, g^{-1}]\cap P$. In the general case we describe a procedure/algorithm to compute a set of generators for $Sat_g(S)$ which terminates if and only if it is finitely generated. Then we consider the more interesting case when $S$ is graded. In particular, if $S$ is graded by a positive matrix $W$ and $g$ is an indeterminate, we show that if we choose a term ordering $\sigma$ of $g$-DegRev type compatible with $W$, then the two operations of computing a $\sigma$-SAGBI basis of $S$ and saturating $S$ with respect to $g$ commute. This fact opens the doors to nice algorithms for the computation of $Sat_g(S)$. In particular, under special assumptions on the grading one can use the truncation of a $\sigma$-SAGBI basis and get the desired result. Notably, this technique can be applied to the problem of directly computing some $U$-invariants, classically called semi-invariants, even in the case that $K$ is not the field of complex numbers.