Bialostocki, Arie; Tran Dinh Luong An analogue of the Erdős-Ginzburg-Ziv theorem for quadratic symmetric polynomials. (English) Zbl 1230.11021 Integers 9, No. 5, 459-465, A36 (2009). 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 \(\mathbb Z_p[x_1, x_2,\dots , x_p]\). Cited in 5 Documents MSC: 11B50 Sequences (mod \(m\)) 11T06 Polynomials over finite fields 13M10 Polynomials and finite commutative rings Keywords:Erdős-Ginzburg-Ziv theorem; symmetric polynomials PDFBibTeX XMLCite \textit{A. Bialostocki} and \textit{Tran Dinh Luong}, Integers 9, No. 5, 459--465, A36 (2009; Zbl 1230.11021) Full Text: DOI EuDML