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 $\overline M=M\otimes_S S[x_{ n+1}]$.
Categories: math.AC