On regular sequences of binomials

Abstract:

Let F ₁,…,F _r denote generic polynomials, i.e. polynomials with indeterminate coefficients. In our treatment of such polynomials and their specializations at the Trieste Workshop in 1992 we determined under which conditions F ₁,…,F _r generate a prime ideal. This is done algebraically in terms of regular sequences and combinatorially in terms of the involved monomials. If these conditions are fulfilled, the sequence F ₁,…,F _r is now said to be strictly admissible.

In this paper we attend to sequences of generic binomials in T ₀,…,T _n with . We show in Section 2 that strict admissiblility for binomials can also be characterized via {\it distinguished} sequences of exponents, a combinatorial concept due to Ch.~Delorme. The main tools of the proofs are developed in Section 3 using methods of graph theory.

In Section 1 connections to monoid theory are outlined: The generic binomials\linebreak F ₁,…,F _r specialize to the binomials , which define the monoid algebra , K any field, of a quotient monoid W of . The strict admissibility of F ₁,…,F _r is closely related to the property of the monoid W to be cancellative and a complete intersection. This generalizes the description of complete intersection monoids given by Ch. Delorme in 1976.

In Section 4 we discuss computational aspects and show in particular that strict admissibility for binomial sequences is decidable in polynomial time.