Saturations of subalgebras, SAGBI bases, and U-invariants

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