Related papers: Asymptotics of Chebyshev Polynomials, V. Residual …

Let $f_1,\dots,f_k\in\mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant…

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…

Let p(t) be a trigonometric polynomial, non-negative on the unit circle. We say that a measure \sigma belongs to a polynomial Szego class, if the logarithm of its density is summable over the circle with the weight p(t). For the associated…

This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We…

Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and…

Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of…

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input…

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the…

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in…

Starting from the orthogonal polynomial expansion of a function $F$ corresponding to a finite positive Borel measure with infinite compact support, we study the asymptotic behavior of certain associated rational functions…

We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial…

We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is a polynomial of degree $j$) under the restriction that the sup-norm of $\sum_{j=1}^m…

We obtain an explicit upper bound on the size of the coefficients of the elliptic modular polynomials $\Phi_N$ for any $N\geq1$. These polynomials vanish at pairs of $j$-invariants of elliptic curves linked by cyclic isogenies of degree…

In the present paper we consider $F_k(x)=x^{k}-\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree…

The purpose of this note is to extend in a simple and unified way some results on orthogonal polynomials with respect to the weight function $$\frac{|T_m(x)|^p}{\sqrt{1-x^2}}\;,\quad-1<x<1\;,$$ where $T_m$ is the Chebyshev polynomial of the…

By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $\alpha\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence…

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the $a_i$ and $b$ are fixed coefficients, and $h$ is an arbitrary integer…

Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…

We derive bulk asymptotics of skew-orthogonal polynomials (sop) $\pi^{\bt}_{m}$, $\beta=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ there exists an $\epsilon >…

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…