Related papers: Quaternionic Generalization of the Enestr\"{o}m-Ka…

In combinatorics, P\'{o}lya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of P\'{o}lya's…

In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients.…

In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the…

We show that all the zeros of the quadrinomial $p(z)=1+\kappa(z+z^{N-1})+z^N$ lie on the unit circle if and only if the inequalities \[ -1\le\kappa\le 1\; (\mbox{ if $N$ is even}),\;\; -1\le\kappa\le N/(N-2)\; (\mbox{ if $N$ is odd}) \]…

We study distribution of zeros of a complex polynomial whose coefficients has been modified. We give a new proof of the theorem of Rubinstein, and with similar method we prove a new theorem that is not generalization of the previous…

Given a polynomial $\sum_\nu a_\nu X^\nu$ of degree $<d$, bounded by one on the unit disk, how large can $\lvert a_0+a_1+\ldots+a_n \rvert$ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D.…

Let $m\geq 2$ be a positive integer. Given a set $E(\omega )\subseteq \mathbb{N}$ we define $r_{N}^{(m)}(\omega )$ to be the number of ways to represent $N\in \mathbb{Z}$ as any combination of sums $\textit{ and }$ differences of $m$…

A Lee-Yang polynomial $ p(z_{1},\ldots,z_{n}) $ is a polynomial that has no zeros in the polydisc $ \mathbb{D}^{n} $ and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}})^{n} $. We show that any real-rooted exponential polynomial of…

We prove two generalisations of the Binomial theorem that are also generalisations of the q-binomial theorem. These generalisations arise from the commutation relations satisfied by the components of the co-multiplications of non-simple…

We discuss a form of a well-known problem of Kakeya for complex polynomials. Let p(z) be a complex polynomial. This problem requires to find disc that contains n zeros of some derivative of p(z), provided that location of several zeros of…

We give a simple proof of a classical theorem by A.M\'at\'e, P.Nevai, and V.Totik on asymptotic behavior of orthogonal polynomials on the unit circle. It is based on a new real-variable approach involving an entropy estimate for the…

We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each…

We prove a general multidimensional invariance principle for a family of U-statistics based on freely independent non-commutative random variables of the type $U_n(S)$, where $U_n(x)$ is the $n$-th Chebyshev polynomial and $S$ is a standard…

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$…

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient…

The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article,…

We generalize Coincidence theorem due to Walsh for symmetric and linear polynomial in n complex variables, that is linear in each of them having total degre n. We discuss case when total degree is smaller then n. This case has been already…

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…