On the Hilbert depth of the Hilbert function of a finitely generated graded module

TL;DR

This paper investigates the Hilbert depth of the Hilbert function of finitely generated graded modules over standard graded algebras, providing explicit calculations and inequalities using hypergeometric functions.

Contribution

It introduces new results on the Hilbert depth for polynomial rings and complete intersection monomial ideals, and establishes inequalities for modules extended by polynomial variables.

Findings

Hilbert depth of polynomial rings equals the number of variables

Complete intersection monomial ideals have maximal Hilbert depth equal to the number of variables

Hilbert depth does not decrease under module extension by polynomial variables

Abstract

Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq k} (-1)^{k-j} \binom{d-j}{k-j} h_{M}(j) \geq 0 \text{ for all } k\leq d\}, $$ where $h_M(-)$ is the Hilbert function of $M$, and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that $\operatorname{hdepth}(h_S)=n$, where $S=K[x_1,\ldots,x_n]$. We show that $\operatorname{hdepth}(h_{S/J})=n$, if $J=(f_1,\ldots,f_d)\subset S$ is a complete intersection monomial ideal with $deg(f_i)\geq 2$ for all $1\leq i\leq d$. Also, we show that $\operatorname{hdepth}(h_{\overline M})\geq \operatorname{hdepth}(h_M)$ for any finitely generated graded $S$-module $M$, where $\overline M=M\otimes_S S[x_{n+1}]$.