Value distribution of elementary symmetric polynomials and its\n perturbations over finite fields

2019/08/30 by Luis A. Medina, Medina, Luis A., L. Brehsner Sepúlveda +3
Computer Science · Mathematics · #05E05 #11T23 #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1908.11793

openalex publication_date 2019/08/30 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

In this article we establish the asymptotic behavior of generating functions\nrelated to the exponential sum over finite fields of elementary symmetric\nfunctions and their perturbations. This asymptotic behavior allows us to\ncalculate the probability generating function of the probability that the the\nelementary symmetric polynomial of degree k and its perturbations returns\n\β \∈ mathbbF_q where mathbbF_q represents the field of q\nelements. Our study extends many of the results known for perturbations over\nthe binary field to any finite field. In particular, we establish when a\nparticular perturbation is asymptotically balanced over a prime field and\nprovide a construction to find such perturbations over any finite field.\n

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