On the Hilbert depth of quadratic and cubic functions

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