Polynomial Equations in Subgroups and Applications

2020/05/11 by Sergeĭ Konyagin, Konyagin, Sergei V., Igor E. Shparlinski +3
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2005.05315

openalex publication_date 2020/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain a new bound for the number of solutions to polynomial equations in cosets of multiplicative subgroups in finite fields, which generalises previous results of P. Corvaja and U. Zannier (2013). We also obtain a conditional improvement of recent results of J. Bourgain, A. Gamburd and P. Sarnak (2016) and S. V. Konyagin, S. V. Makarychev, I. E. Shparlinski and I. V. Vyugin (2019) on the structure of solutions to the reduction of the Markoff equation x² + y² + z² = 3 x yz modulo a prime p.

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