5 papers
On some polynomial version on the sum-product problem for subgroups
Sofia Aleshina, Ilya Vyugin
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 polynomi…
Polynomial Equations in Subgroups and Applications
Sergei V. Konyagin, Igor E. Shparlinski, Ilya V. Vyugin
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 a…
On the Bound of Inverse Images of a Polynomial Map
Ilya Vyugin
Let be some polynomials. The upper bound on the number of such that are roots of unit of order is obtained. This…
On the Riemann-Hilbert Problem for Difference and -Difference Systems
Ilya Vyugin, Roman Levin
In this paper we study an analogue of the classical Riemann-Hilbert problem stated for the classes of difference and -difference systems. The Birkhoff's existence theorem was ge…
Expansions for solutions of the Schlesinger equation at a singular point
Ilya Vyugin
A local behavior of solutions of the Schlesinger equation is studied. We obtain expansions for this solutions, which converge in some neighborhood of a singular point. As a corolla…