Saturations of Subalgebras, SAGBI Bases, and U-invariants

Abstract 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 Sat g ( S ) = S [ g , g − 1 ] ∩ 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 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 Sat g ( 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.