An Analogue of the Erdős-ginzburg-ziv Theorem for Quadratic Symmetric Polynomials
نویسندگان
چکیده
Let p be a prime and let φ ∈ Zp[x1, x2, . . . , xp] be a symmetric polynomial, where Zp is the field of p elements. A sequence T in Zp of length p is called a φ-zero sequence if φ(T ) = 0; a sequence in Zp is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Define g(φ, Zp) to be the smallest integer l such that every sequence in Zp of length l contains a φ-zero sequence; if l does not exist, we set g(φ, Zp) = ∞. Define M(φ, Zp) to be the set of all φ-zero free sequences of length g(φ, Zp)−1, whenever g(φ, Zp) is finite. The aim of this paper is to determine the value of g(φ, Zp) and to describe the set M(φ, Zp) for a quadratic symmetric polynomial φ in Zp[x1, x2, . . . , xp].
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