Abstract
We obtain a new bound for the number of solutions to polynomial equations in cosets of multiplicative subgroups in finite fields, which generalizes 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 = 3xyz modulo a prime p.
Dedicated to the Memory of Jean Bourgain
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Acknowledgements
The authors are very grateful to William Chen for clarifying discussion concerning his work [6] and to the referee for the careful reading of the manuscript and valuable remarks.
This work was supported by the Australian Research Council Grants DP180100201 and DP200100355 (Shparlinski) and by the Russian Science Foundation Grant 19-11-00001 (Vyugin).
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Konyagin, S.V., Shparlinski, I.E., Vyugin, I.V. (2022). Polynomial Equations in Subgroups and Applications. In: Avila, A., Rassias, M.T., Sinai, Y. (eds) Analysis at Large. Springer, Cham. https://doi.org/10.1007/978-3-031-05331-3_12
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