Solutions of polynomial equation over mathbbF_p and new bounds of\n additive energy

2015/03/26 by Il'ya Vladimirovich Vyugin, Vyugin, Ilya, Sergey Makarychev +1
Mathematics · #Meromorphic and Entire Functions #Analytic Number Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.1504.01354

Abstract

We present a new proof of Corvaja and Zannier's citeC-Z the upper bound of\nthe number of solutions (x,y) of the algebraic equation P(x,y)=0 over a\nfield mathbbF_p (p is a prime), in the case, where x\∈ g₁G, y\∈\ng₂G, (g₁G, g₂G -- are cosets by some subgroup G of a multiplicative\ngroup mathbbF_p^*). The estimate of Corvaja and Zannier was improved in\naverage, and some applications of it has been obtained. In particular we\npresent the new bounds of additive and polynomial energy.\n

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