Saturations of subalgebras, SAGBI bases, and U-invariants

Given a polynomial ring P over a field K, an element g∈P, and a K-subalgebra S of P, we deal with the problem of saturating S with respect to g, i.e. computing Satg(S)=S[g,g−1]∩P. In the general case we describe a procedure/algorithm to compute a set of generators for Satg(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 σ of g-DegRev type compatible with W, then the two operations of computing a σ-SAGBI basis of S and saturating S with respect to g commute. This fact opens the doors to nice algorithms for the computation of Satg(S). In particular, under special assumptions on the grading one can use the truncation of a σ-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.