Related papers: Two-point polynomial patterns in subsets of positi…

Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with applications to John-type approximation and sets with small doubling. First we show that if $A \subset \mathbb{Z}$ with $|A+A| \le…

We establish the following quantitative form of the Green--Tao theorem: if a set $\mathcal{A}$ of relative density $\delta$ within the primes up to $N$ contains no nontrivial arithmetic progressions of length $k\geq 4$, then $\delta\ll…

In this article, we study some special cases of the problem of classifying polynomials $p:\mathbb{R}^2_+\to (0,\infty)$ for which the net $\{\frac{1}{p(m,n)}\}_{m,n\in \mathbb{Z}_+}$ is a completely monotone net, where $p(x,y)=b(x)+a(x)y$,…

The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real…

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative…

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the…

For nonnegative real numbers $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\alpha+\beta+\gamma >0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C…

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…

The polynomial Szemer\'{e}di theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of…

Let $s(n)$ denote the number of ones in the binary expansion of the nonnegative integer $n$. How does $s$ behave under addition of a constant $t$? In order to study the differences \[s(n+t)-s(n),\] for all $n\ge0$, we consider the…

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r)…

Let \( G \) be a graph of order \( n \) with maximum degree $\Delta$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study…

We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given where each…

We introduce the class of {\em strongly Rayleigh} probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal…

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^\alpha e^{-x}$ the gamma density and $w_2(x) = x^\alpha…

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in…

Let $ V_{n} = X_{1,n} + X_{2,n} + \cdots + X_{n,n}$ where $X_{i,n}$ are Bernoulli random variables which take the value $1$ with probability $b(i;n)$. Let $\lambda_{n} = \sum\limits_{i=1}^{n} b(i;n) $, $\lambda = \lim\limits_{n \to \infty}…

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…

We consider polynomials $p_n^{\omega}(x)$ that are orthogonal with respect to the oscillatory weight $w(x)=e^{i\omega x}$ on $[-1,1]$, where $\omega>0$ is a real parameter. A first analysis of $p_n^{\omega}(x)$ for large values of $\omega$…

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards…