On the arithmetic quasi depth

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 quasi depth of $h$ by $qdepth(h)=\max\{d\;:\; \sum_{j\leq k} (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$. We show that $qdepth(h)$ is a natural generalization for the quasi depth of a subposet $P\subset 2^{[n]}$ and we prove some basic properties of it. Given $h(j)=aj^p+b$, $j\geq 0$, with $a,b,n$ positive integers, we compute $qdepth(h)$ for $n=1,2$ and we give sharp bounds for $qdepth(h)$ for $p\geq 3$. Also, for $h(j)=a_nj^n+\cdots+a_1j+a_0$, $j\geq 0$, with $a_i\geq 0$, we prove that $qdepth(h)\leq 2^{n+1}$.Comment: 18 page