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Sequences of discrete random variables are studied whose probability generating functions are zero-free in a sector of the complex plane around the positive real axis. Sharp bounds on the cumulants of all orders are stated, leading to…
We investigate three types of averaging principles and the normal deviation for multi-scale stochastic differential equations (in short, SDEs) with polynomial nonlinearity. More specifically, we first demonstrate the strong convergence of…
We establish necessary and sufficient conditions for an arbitrary polynomial of degree $n$, especially with only real roots, to be trivial, i.e. to have the form a(x-b)^n. To do this, we derive new properties of polynomials and their roots.…
The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at…
We formulate a conjecture concerning spectral factorization of a class of trigonometric polynomials of two variables and prove it for special cases. Our method uses relations between the distribution of values of a polynomial of two…
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…
The behavior under iteration of the critical points of polynomial maps plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory,…
We discuss some extensions and refinements of the variance bounds for both real and complex numbers. The related bounds for the eigenvalues and spread of a matrix are also derived here.
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
In this paper, we consider a problem of Lang about finiteness of torsion points on plane rational curves over $\mathbb C$, and prove some results towards a matrix analogue of this problem.
Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation,…
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…
The aim of this paper is twofold. The first part is concerned with the associated and the so-called co-polynomials, i.e. new sequences obtained when finite perturbations of the recurrence coefficients are considered. In the second part we…
We survey Vojta's higher-dimensional generalizations of the $abc$ conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "$abcd$ conjecture" implies a…
The Gauss--Lucas and B\^{o}cher--Grace--Marden theorems are classical results in the geometry of polynomials. Proofs of the these results are available in the literature, but the approaches are seemingly different. In this work, we show…
We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel…
We consider Markov chains which are polynomially mixing, in a weak sense expressed in terms of the space of functions on which the mixing speed is controlled. In this context, we prove polynomial large and moderate deviations inequalities.…
The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and…
Special polynomials associated with rational solutions of the second Painlev\'{e} equation and other members of its hierarchy are discussed. New approach, which allows one to construct each polynomial is presented. The structure of the…