arXiv subjects
Publications and source records attributed to Ilya V. Vyugin.
We prove that almost all solutions of the Markoff-Hurwitz equation over a residue field modulo $p$ can be obtained from one another by a chain of natural transformations. We also study recurrence sequences considered modulo prime $p$.
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 $x^2 + y^2 + z^2 = 3 x yz$ modulo a prime $p$.
The paper is devoted to some applications of Stepanov method. In the first part of the paper we obtain the estimate of the cardinality of the set, which is obtained as an intersection of additive shifts of some different subgroups of F^*_p. In the second part we prove a new upper bound for Heilbronn's exponential sum and obtain a series of applications of our result to distribution of Fermat quotients. Also we study additive decompositions of multiplicative subgroups.