Binomial Ideals David Eisenbud and Bernd Sturmfels

Introduction. It is notoriously difficult to deduce anything about the structure of an ideal or scheme by directly examining its defining polynomials. A notable exception is that of monomial ideals. Combined with techniques for making fiat degenerations of arbitrary ideals into monomial ideals (typically, using Gr6bner bases), the theory of monomial ideals becomes a useful tool for studying general ideals. Any monomial ideal defines a scheme whose components are coordinate planes. These objects have provided a useful medium for exchanging information between commutative algebra, algebraic geometry, and combinatorics. This paper initiates the study of a larger class of ideals whose structure can still be interpreted directly from their generators: binomial ideals. By a binomial in a polynomial ring S k[xl,..., Xn], we mean a polynomial with at most two terms, say axe+ bx, where a,b k and , fl Z_. We define a binomial ideal to be an ideal of S generated by binomials, and a binomial scheme (or binomial variety, or binomial algebra) to be a scheme (or variety or algebra) defined by a binomial ideal. For example, it is well known that the ideal of algebraic relations on a set of monomials is a prime binomial ideal (Corollary 1.3). In Corollary 2.6 we shall see that every binomial prime ideal has essentially this form. A first hint that there is something special about binomial ideals is given by the following result, a weak form of what is proved below (see Corollary 2.6 and Theorem 6.1).