7 citations · 16 across the 11 of their papers we have counts for
11 papers
On the structure of solutions graph of the Markoff-Hurwitz equation and recurrence sequences
Ilya V. Vyugin
We prove that almost all solutions of the Markoff-Hurwitz equation over a residue field modulo can be obtained from one another by a chain of natural transformations. We also s…
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 new bound for the number of solutions of polynomial equations in subgroups and the structure of graphs of Markoff triples
S. V. Konyagin, S. V. Makarychev, I. E. Shparlinski +1
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 componen…
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…
Solutions of polynomial equation over and new bounds of additive energy
Ilya Vyugin, Sergey Makarychev
We present a new proof of Corvaja and Zannier's \cite{C-Z} the upper bound of the number of solutions of the algebraic equation over a field (…
Solvability of linear differential systems in the Liouvillian sense
Renat Gontsov, Ilya Vyugin
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular sing…
Intersections of multiplicative subgroups and Heilbronn's exponential sum
Ilya D. Shkredov, Elena V. Solodkova, Ilya V. Vyugin
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 intersect…
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…
Apparent singularities of Fuchsian equations, and the Painlevé VI equation and Garnier systems
R. R. Gontsov, I. V. Vyugin
We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions fo…