On the Hilbert depth of quadratic and cubic functions

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 $\operatorname{hdepth}(h_3)\leq 67$.