The cones of Hilbert functions of squarefree modules (Q2904278)

In the paper under review, the authors focus on different possible definitions of squarefreeness for modules over the polynomial ring \(S = \mathbb{K}[x_1, \dots, x_n]\) with the standard \(\mathbb{N}^n\) grading. They investigate the cone of Hilbert function of squarefree modules and determine both the extremal rays and the defining inequalities of the cone of Hilbert functions of these modules. Then, they restrict to the class of squarefree modules generated in degree zero. This case can be reduced to Hilbert functions of Stanley-Reisner rings using Gröbner bases. Again, the authors describe the extremal rays and defining inequalities of the cone of Hilbert functions of these modules.NEWLINENEWLINEThe defining inequalities in this last case give a linear bound on the growth of the Hilbert function of a Stanley-Reisner ring. Finally the authors compare this bound to the non-linear but optimal bound given by the Kruskal-Katona Theorem.