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On some polynomial version on the sum-product problem for subgroups

We generalize two results about subgroups of multiplicative group of finite field of prime order. In particular, the lower bound on the cardinality of the set of values of polynomial $P(x,y)$ is obtained under the certain conditions, if variables $x$

Externí odkaz: http://arxiv.org/abs/2008.08684

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Polynomial Equations in Subgroups and Applications

Autor: Konyagin, Sergei V., Shparlinski, Igor E., Vyugin, Ilya V.

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 rece

Externí odkaz: http://arxiv.org/abs/2005.05315

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Dispersion of the Arnold's asymptotic ergodic Hopf invariant and a formula for its calculation

Autor: Akhmet'ev, P. M., Vyugin, I. V.

The paper contains an example to describe magnetic fields in a conductive medium. The authors assume that new applications for turbulent magnetic fields in the case the magnetic field is not left- and right- polarized are possible.

Externí odkaz: http://arxiv.org/abs/1906.12131

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Akademický článek

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On the Bound of Inverse Images of a Polynomial Map

Autor: Vyugin, Ilya

Let $f_1(x),\ldots,f_n(x)$ be some polynomials. The upper bound on the number of $x\in\mathbb F_p$ such that $f_1(x),\ldots,f_n(x)$ are roots of unit of order $t$ is obtained. This bound generalize the bound of the paper \cite{V-S} to the case of pol

Externí odkaz: http://arxiv.org/abs/1811.08930

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On the new bound for the number of solutions of polynomial equations in subgroups and the structure of graphs of Markoff triples

Autor: Konyagin, S. V., Makarychev, S. V., Shparlinski, I. E., Vyugin, I. V.

We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the "giant component" and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo

Externí odkaz: http://arxiv.org/abs/1711.05335

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On the Riemann-Hilbert Problem for Difference and $q$-Difference Systems

Autor: Vyugin, Ilya, Levin, Roman

In this paper we study an analogue of the classical Riemann-Hilbert problem stated for the classes of difference and $q$-difference systems. The Birkhoff's existence theorem was generalized in this paper.

Externí odkaz: http://arxiv.org/abs/1702.08323

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