SAGBI and SAGBI-Gröbner bases over principal ideal domains
SAGBI bases (``Subalgebra Analogues of Gröbner Bases for Ideals'') were introduced by \textit{L. Robbiano} and \textit{M. Sweedler} [in: Commutative Algebra, Proc. Workshop, Salvador 1988, Lect. Notes Math. 1430, 61-87 (1988; Zbl 0725.13013)]. They are defined as follows: Let \(A:=R[f_1,\dots,f_m]\subseteq R[x_1,\dots,x_n]\) be the algebra generated over \(R\) by the polynomials \(f_1,\dots f_m\) and assume a term-order is fixed in \(R[x_1,\dots,x_n]\). Then a subset \(S\) of \(A\) is a SAGBI basis for \(A\) if \(R[Lt(S)]=R[Lt(A)]\) (where \(Lt(S)\) and \(Lt(A)\) denote the set of leading terms of the elements of \(S\) and of \(A\) respectively). The above definition is usually given in the case where \(R\) is a field, but in the present paper, the authors are mainly interested in the more general case in which \(R\) is a principal ideal domain. First of all they discuss a Buchberger-type algorithm for the computation of SAGBI bases (when SAGBI bases exist, which is not always the case). Successively they study SAGBI-Gröbner bases for ideals of \(A\) (which are bases with properties similar to Gröbner bases of ideals in polynomials rings). Assuming that \(A\) has a finite SAGBI bases, they give an algorithm for computing SAGBI-Gröbner bases. In the last section they show that strong SAGBI bases -- the analogue of strong Gröbner bases for ideals in polynomial rings -- always exist (under the assumption that a finite SAGBI basis exists) and can be computed with an algorithm. This construction furthermore implies some results in integer programming.
scientific article; zbMATH DE number 1271791