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On the Hilbert depth of quadratic and cubic functions

Mircea Cimpoeas, Silviu Balanescu

Abstract

Given a numerical function $h:\mathbb Z_{\geq 0}\to\mathbb Z_{\geq 0}$ with $h(0)>0$, the Hilbert depth of $h$ is $\operatorname{hdepth}(h)=\max\{d\;:\;\sum\limits_{j=0}^k (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$; see arXiv:2309.10521 . In this note, we study the Hilbert depth of the functions $h_2(j)=aj^2+bj+e$, $j\geq 0$, and $h_3(j)=aj^3+bj^2+cj+e$, $j\geq 0$, where $a,b,c,e$ are some integers with $a,e>0$. We prove that if $b<0$ and $b^2\leq 4ae$ then $\operatorname{hdepth}(h_2)\leq 11$, and, if $b<0$and $b^2>4ae$ then $\operatorname{hdepth}(h_2)\leq 13$. Also, we show that if $b<0$ and $b^2\leq 3ac$ then $\operatorname{hdepth}(h_3)\leq 67$.

On the Hilbert depth of quadratic and cubic functions

Abstract

Given a numerical function with , the Hilbert depth of is ; see arXiv:2309.10521 . In this note, we study the Hilbert depth of the functions , , and , , where are some integers with . We prove that if and then , and, if and then . Also, we show that if and then .
Paper Structure (3 sections, 14 theorems, 56 equations, 1 figure)

This paper contains 3 sections, 14 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The quadratic function
  3. The cubic function

Key Result

Theorem 1.1

(see lucrare7) Let $h\in\mathcal{H}_0$, $h(j)=a_nj^n+\cdots+a_1j+a_0,\; j\geq 0$, such that $a_i\geq 0$ for all $1\leq i\leq n$ and $a_0>0$. Then

Figures (1)

  • Figure 1: The domain $K$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Conjecture 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • ...and 21 more