The classical Erdős–Ginzburg–Ziv constant of a group G denotes the smallest positive integer \(\ell \) such that any sequence S of length at least \(\ell \) contains a zero-sum subsequence of length \(\exp (G).\) In the recent paper (Integers 22: Paper No. A102, 17 pp., 2022), 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 the 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 \(k=2, 3,4,5.\)

群G的经典 Erdős–Ginzburg–Ziv 常数表示最小正整数\(\ell \)，使得任何长度至少为\(\ell \)的序列S包含长度为\(\exp)的零和子序列(G).\)在最近的论文（Integers 22: Paper No. A102, 17 pp., 2022）中，Caro 和 Schmitt 推广了这个概念，使用 m 次对称多项式\(e_m(S)\)代替S的元素之和并考虑给定长度t的子序列。特别是，他们定义了更高阶的 Erdős–Ginzburg–Ziv 常数EGZ ( t ,  R,  m ) 的有限交换环R并给出了这些常数的几个下限和上限。本文的目的是在\(R={\mathbb {F}}_q^{n}.\) 的情况下提供EGZ ( t ,  R ,  m )的下界和上限。\)这里给出的下界已分别获得，使用 Lovász 局部引理和清除方法，对于足够大的n，他们击败了 Caro 和 Schmitt 为同类环提供的下界。最后，我们证明了从 Ellenberg-Gijswijt 和 Sauermann 结果得出的上限集问题的封闭形式上限，假设\(q = p^k,\) \(t = p,\) 和\(m=p-1.\)此外，使用切片排序方法，我们推导出一个凸优化问题，为\(q = 3^k,\) \(t = 3,\) \提供最佳边界(m=2,\)和\(k=2, 3,4,5.\)