On the arithmetic Hilbert depth

Let $h:\mathbb{Z}\to {\mathbb{Z}}_{\ge 0}$ be a nonzero function with $h(k)=0$ for $k\ll 0$. We define the Hilbert depth of $h$ by $\mathrm{hdepth}(h)=\max \{d:{\sum }_{j\le k}(-1{)}^{k-j}\left(\genfrac{}{}{0px}{}{d-j}{k-j}\right)h(j)\ge 0\text{ for all }k\le d\}$. We show that $\mathrm{hdepth}(h)$ is a natural generalization for the Hilbert depth of a subposet $\mathrm{P}\subset 2^{[n]}$ and we prove some basic properties of it. Given $h(j)=\{\begin{array}{ll}a{j}^n+b,& j\ge 0\\ 0,& j<0\end{array}$, with $a,b,n$ positive integers, we compute $\mathrm{hdepth}(h)$ for $n=1,2$ and we give upper bounds for $\mathrm{hdepth}(h)$ for $n\ge 3$. More generally, if $h(j)=\{\begin{array}{ll}P(j),& j\ge 0\\ 0,& j<0\end{array}$, where $P(j)$ is a polynomial of degree $n$, with non-negative integer coefficients, and $P(0)>0$, we show that $\mathrm{hdepth}(h)\le 2^{n+1}$.