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Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…

The celebrated Ore-DeMillo-Lipton-Schwartz-Zippel (ODLSZ) lemma asserts that n-variate non-zero polynomial functions of degree d over a field $\mathbb{F}$ are non-zero over any "grid" $S^n$ for finite subset $S \subseteq \mathbb{F}$, with…

Brannan showed that a normalized univalent polynomial of the form $P(z)=z+a_2 z^2+\ldots + a_{n-1}z^{n-1}+\frac{z^n}{n}$ is starlike if and only if $a_2=\ldots=a_{n-1}=0$. We give a new and simple proof of his result, showing further that…

Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We…

Let $P(z)$ be a polynomial of degree $n$ having no zero in $|z|<k$ where $k\geq 1,$ then for every real or complex number $\alpha$ with $|\alpha|\geq 1$ it is known \begin{equation*} \underset{|z|=1}{\max}|D_\alpha P(z)|\leq…

We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that…

Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant…

Given two polynomials $P(\underline x)$, $Q(\underline x)$ in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values $P(\underline n), Q(\underline n)$ at integer points…

We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to…

While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq…

Let $D_\alpha P(z)=nP(z)+(\alpha-z)P^{\prime}(z)$ denote the polar derivative of a polynomial $P(z)$ of degree $n$ with respect to a point $\alpha\in\mathbb{C}.$ In this paper, we present a correct proof, independent of Laguerre's theorem,…

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 a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$…

In the geometry of polynomials, Schoenberg's conjecture, now a theorem, is a quadratic inequality between the zeros and critical points of a polynomial whose zeros have their centroid at the origin. We call its generalizations to other…

In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…

For odd $N\geq 5$, we establish a short exact sequence about motivic double zeta values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. From this we classify all the relations among depth-graded motivic double zeta values…

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of…

We compute the special values of partial zeta function at $s=0$ for family of real quadratic fields $K_n$ and ray class ideals $\fb_n$ such that $\fb_n^{-1} = [1,\delta(n)]$ where the continued fraction expansion of $\delta(n)$ is purely…

If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left\|a_nz+\frac{a_s}{\binom{n}{s}}\right\|_{p}\leq…

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d, d\geq 1$, unbounded in both degree and genus. By using a closed formula for the Mahler measure introduced in "Volume function and Mahler…