Saturations of subalgebras, SAGBI bases, and U-invariants. (March 2022)

• Record Type: Journal Article
• Authors: Bigatti, Anna Maria; Robbiano, Lorenzo
• Is Part Of: Journal of symbolic computation. Volume 109(2022)
• Page Start: 259
• Page End: 282
• Publication Date: 2022-03
• Subjects: 13P10 -- 08A30 -- 13-04 -- 14R20 -- 68W30; Subalgebra saturation -- SAGBI bases -- CoCoA -- U-invariants; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.07.006
• Languages: English
• ISSNs: 0747-7171

Abstract

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.