On the arithmetic Hilbert depth

TL;DR

This paper introduces a generalized concept of Hilbert depth for functions on integers, explores its properties, and computes bounds for specific polynomial-based functions, extending the notion to subposets of the Boolean lattice.

Contribution

It defines a new generalized Hilbert depth for integer-valued functions and establishes its basic properties and bounds, extending previous combinatorial and algebraic concepts.

Findings

Computed Hilbert depth for linear and quadratic functions.

Established upper bounds for polynomial functions of degree n.

Linked Hilbert depth to subposets of the Boolean lattice.

Abstract

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