On Hilbertian ideals

Abstract

Abhyankar defined the index of a monomial in a matrix of indeterminates X to be the maximal size of any minor of X whose principal diagonal divides the given monomial. Using this concept, he characterized a free basis for general type of determinantal ideals formed by the minors coming from a saturated subset of X. In this paper, to a monomial in X of index p we associate a combinatorial object called a superskeleton of latitude p, which can loosely be described as a p-tuple of "almost nonintersecting paths" in a rectangular lattice of points. Using this map, we prove that the ideal generated by the p by p minors of a saturated set in X is hilbertian, i.e., the Hilbert polynomial of this ideal coincides with its Hilbert function for all nonnegative integers.