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An analogue of the Erdős-Ginzburg-Ziv theorem for quadratic symmetric polynomials. (English) Zbl 1230.11021

Summary: Let \(p\) be a prime and let \(\varphi \in \mathbb Z_p[x_1, x_2, \dots, x_p]\) 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 \(\varphi \)-zero sequence if \(\varphi(T) = 0\); a sequence in \(\mathbb Z_p\) is called a \(\varphi \)-zero free sequence if it does not contain any \(\varphi \)-zero subsequence. Define \(g(\varphi, \mathbb Z_p)\) to be the smallest integer \(l\) such that every sequence in \(\mathbb Z_p\) of length \(l\) contains a \(\varphi\)-zero sequence; if \(l\) does not exist, we set \(g(\varphi, \mathbb Z_p) = \infty\). Define \(M(\varphi, \mathbb Z_p)\) to be the set of all \(\varphi \)-zero free sequences of length \(g(\varphi, \mathbb Z_p) - 1\), whenever \(g(\varphi, \mathbb Z_p)\) is finite. The aim of this paper is to determine the value of \(g(\varphi, \mathbb Z_p)\) and to describe the set \(M(\varphi, \mathbb Z_p)\) for a quadratic symmetric polynomial \(\varphi\) in \(\mathbb Z_p[x_1, x_2,\dots , x_p]\).

MSC:

11B50 Sequences (mod \(m\))
11T06 Polynomials over finite fields
13M10 Polynomials and finite commutative rings
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