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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2309.10521 |
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| _version_ | 1866909110129131520 |
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| author | Balanescu, Silviu Cimpoeas, Mircea |
| author_facet | Balanescu, Silviu Cimpoeas, Mircea |
| contents | 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 $\operatorname{hdepth}(h)\leq 2^{n+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10521 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the arithmetic Hilbert depth Balanescu, Silviu Cimpoeas, Mircea Number Theory Combinatorics 05A18, 05A20, 06A07, 13D40 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 $\operatorname{hdepth}(h)\leq 2^{n+1}$. |
| title | On the arithmetic Hilbert depth |
| topic | Number Theory Combinatorics 05A18, 05A20, 06A07, 13D40 |
| url | https://arxiv.org/abs/2309.10521 |