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Solutions of Polynomial Equations in Subgroups of \(\mathbb {F}_p^*\)

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Abstract

We present an upper bound on the number of solutions of an algebraic equation \(P(x,y)=0\) where x and y belong to the union of cosets of some subgroup of the multiplicative group \(\kappa ^*\) of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927–1942, 2013) to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy.

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Notes

  1. [x]—the integer part of x.

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Acknowledgements

The authors are grateful to Sergei Konyagin, Ilya Shkredov and Ian Marshall for their attention and useful comments. The authors are particularly grateful to Igor Shparlinski and Umberto Zannier for their contribution to the formulation of the problem, which is considered in the paper.

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Authors and Affiliations

  1. Skolkovo Institute of Science and Technology, National Research University Higher School of Economics, Moscow, Russia

    Sergei Makarychev

  2. Institute for Information Transmission Problems RAS, National Research University Higher School of Economics, Steklov Mathematical Institute RAS, Moscow, Russia

    Ilya Vyugin

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  1. Sergei Makarychev
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  2. Ilya Vyugin
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Correspondence to Ilya Vyugin.

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Dedicated to the 70th anniversary of Rafail Kalmanovich Gordin.

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The work of I.V. Vyugin is supported by the Russian Science Foundation grant RSF 19-11-00001 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.

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Makarychev, S., Vyugin, I. Solutions of Polynomial Equations in Subgroups of \(\mathbb {F}_p^*\). Arnold Math J. 5, 105–121 (2019). https://doi.org/10.1007/s40598-019-00112-z

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  • DOI: https://doi.org/10.1007/s40598-019-00112-z

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