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An Analogue of the Erdős–Ginzburg–Ziv Theorem for Quadratic Symmetric Polynomials
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Arie Bialostocki
Published/Copyright:
November 25, 2009
Abstract
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 inℤp 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].
Keywords.: Erdős–Ginzburg–Ziv; symmetric polynomials
Received: 2008-12-23
Accepted: 2009-05-20
Published Online: 2009-11-25
Published in Print: 2009-November
© de Gruyter 2009
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Articles in the same Issue
- An Analogue of the Erdős–Ginzburg–Ziv Theorem for Quadratic Symmetric Polynomials
- Counting Determinants of Fibonacci–Hessenberg Matrices Using LU Factorizations
- A Short Proof of a Series Evaluation in Terms of Harmonic Numbers
- Generalization of an Identity Involving the Generalized Fibonacci Numbers and Its Applications
- Inductive Methods and Zero-Sum Free Sequences
- On the Number of Zero-Sum Subsequences of Restricted Size
- On the Average Asymptotic Behavior of a Certain Type of Sequence of Integers
- On the Kernel of the Coprime Graph of Integers
- Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation
- Balanced Subset Sums in Dense Sets of Integers
- Some Results for Generalized Harmonic Numbers
