On binomial complete intersections

Abstract

We consider homogeneous binomial ideals I=(f1,…,fn) in K[x1,…,xn], where fi=aixidi−bimi and ai≠0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by xidi for i=1,…,n form a vector space basis for the corresponding quotient, and we describe the Macaulay dual generator in terms of a directed graph that we associate to I. These two properties can be seen as a natural generalization of well-known properties for monomial complete intersections. Moreover, we give a description of the radical of the resultant of I in terms of the directed graph.