Hilbert Depth of Gotzmann Ideals

Abstract

The Hilbert function of a homogeneous ideal is determined by its associated lexsegment ideal, while the Gotzmann property ensures that the ideal and its lexsegment ideal have the same number of minimal generators. For homogeneous monomial ideals with fewer minimal generators than the number of variables in the polynomial ring, the classes of critical monomial ideals and Gotzmann ideals coincide. In this article, we determine the Hilbert depth of this class of monomial ideals. As an application, we compute the Hilbert depth of the edge ideal of a star graph and propose it as an algebraic graph invariant.