Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Abstract

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$, which satisfy the following two conditions: (i) every sequence in $\mathbb {Z}_{p}$ of length 2p−1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p−1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.