Bounds on the Higher Degree Erdős-Ginzburg-Ziv Constants over ${\mathbb{F}}_q^n$

The classical Erdős-Ginzburg-Ziv constant of a group $G$ denotes the smallest positive integer $\mathrm{\ell }$ such that any sequence $S$ of length at least $\mathrm{\ell }$ contains a zero-sum subsequence of length $\exp (G)$. In a recent paper, Caro and Schmitt generalized this concept, using the $m$-th degree symmetric polynomial $e_m(S)$ instead of the sum of the elements of $S$ and considering subsequences of a given length $t$. In particular, they defined the higher degree Erdős-Ginzburg-Ziv constants $EGZ(t,R,m)$ of a finite commutative ring $R$ and presented several lower and upper bounds to these constants. This paper aims to provide lower and upper bounds for $EGZ(t,R,m)$ in case $R={\mathbb{F}}_q^n$. The lower bounds here presented have been obtained, respectively, using Lovász Local Lemma and the Expurgation method and, for sufficiently large $n$, they beat the lower bound provided by Caro and Schmitt for the same kind of rings. Finally, we prove closed form upper bounds derived from the Ellenberg-Gijswijt and Sauermann results for the cap-set problem assuming that $q=p^k$, $t=p$, and $m=p-1$. Moreover, using the Slice Rank method we derive a convex optimization problem that provides the best bounds for $q=3^k$, $t=3$, $m=2$ and $k=2,3,4,5$.