SAGBI Bases Under Composition

Our interest in the subject of this paper is inspired by Hong (1998), where Hoon Hong addresses the problem of the behavior of Gröbner bases under composition of polynomials. More precisely, let Θ be a set of polynomials, as many as the variables in our polynomial ring. The question then is under which conditions on these polynomials it is true that for an arbitrary Gröbner basis G (with respect to some term ordering), the composed set G ◦Θ is also a Gröbner basis (with respect to the same ordering). If this is the case, then we say that the composition commutes with Gröbner bases computation. The main result in Hong (1998) is that this happens if and only if the composition is “compatible” with the ordering and the nondivisibility (see Section 2.2 and Section 4 respectively, for the terminology). An algorithmic approach for studying subalgebras of polynomial rings, analogous to Buchberger’s Gröbner bases theory for ideals, has been introduced by Robbiano and Sweedler (1990), and independently by Kapur and Madlener (1989); the bases hereby constructed are called SAGBI bases (Subalgebra Analog to Gröbner Bases for Ideals). Since many of the basic concepts of Gröbner bases transfer to the subalgebra case, it is natural to ask under which conditions composition by a set Θ commutes with SAGBI bases computation. The main contribution of this paper is to show that this is the case if and only if the composition is compatible with the ordering, i.e. we need only one of the conditions necessary for Gröbner bases. The reason that only this condition is sufficient in the subalgebra case is that it implies the “SAGBI counterpart”, called nonequality below, of the nondivisibility. It follows that commutation with SAGBI bases computation is a weaker condition than commutation with Gröbner bases computation. The natural application of the results in this paper is the same as for Gröbner bases in Hong (1998): composed objects often occur in real-life mathematical models, and given a set F of polynomials in which the variables are defined in terms of other variables, it should be more efficient to compute a SAGBI basis of F before carrying out the composition. (Note, however, that in contrast to Gröbner bases, SAGBI bases computation may