arXiv subjects
Publications and source records attributed to Ilya Vyugin.
We study the $K$-Fibonacci sequence $\mathcal{F}_p$ modulo prime $p$. Cardinalities of sets $|\mathcal{F}_p+\mathcal{F}_p|$ and $|\mathcal{F}_p\cdot\mathcal{F}_p|$ are estimated. We present the method of estimating doubling constant of some $m$-dimensional recurrent sets in $\mathbb{F}_p$.
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 $x$ and $y$ belong to a subgroup $G$ of the multiplicative group of the filed of residues. Also the paper contains a proof of the result that states that if a subgroup $G$ can be presented as a set of values of the polynomial $P(x,y)$, where $x\in A$, and $y\in B$ then the cardinalities of sets $A$ and $B$ are close (in order) to a square root of the cardinality of subgroup $G$.
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 polynomials of degrees greater than one. The bound is obtained over fields of positive characteristic and over the complex field.
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.
We present a new proof of Corvaja and Zannier's \cite{C-Z} the upper bound of the number of solutions $(x,y)$ of the algebraic equation $P(x,y)=0$ over a field $\mathbb{F}_p$ ($p$ is a prime), in the case, where $x\in g_1G$, $y\in g_2G$, ($g_1G$, $g_2G$ -- are cosets by some subgroup $G$ of a multiplicative group $\mathbb{F}_p^*$). The estimate of Corvaja and Zannier was improved in average, and some applications of it has been obtained. In particular we present the new bounds of additive and polynomial energy.
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 singular points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small.
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 corollary the similar result for the sixth Painlevé equation was obtained. In our analysis, we use the isomonodromic approach to solve this problem.