On binomial complete intersections

Abstract

We consider homogeneous binomial ideals $I=(f_1,\dots ,f_n)$ in $K[x_1,\dots ,x_n]$, where $f_i=a_i{x}_i^{d_i}-b_i{m}_i$ and $a_i\ne 0$. When such an ideal is a complete intersection, we show that the monomials which are not divisible by $x_i^{d_i}$ for $i=1,\dots ,n$ 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.

arXiv comment21 pages, 3 figures. v2: Extended results to be valid over any field. To appear in Journal of Algebra