Related papers: On a problem due to Littlewood concerning polynomi…

Polynomials with all the coefficients in $\{ 0,1\}$ and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in $\{ -1,1\}$ are called Littlewood polynomials. By exploiting an algorithm developed…

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…

Let $\mu$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $\alpha_n$ its Verblunsky coefficients, and $\Phi_n$ its monic orthogonal polynomials. In this paper we…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

Expansive polynomials (whose roots are greater than 1 in modulus) often arise in dynamical systems and other computational problems. This paper examines the expansivity gap (the gap between 1 and the smallest modulus of the roots) of these…

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence…

This note is motivated by an old result of Kronecker on monic polynomials with integer coefficients having all their roots in the unit disc. We call such polynomials Kronecker polynomials for short. Let $k(n)$ denote the number of Kronecker…

If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots $\alpha_1$, ..., $\alpha_N$, then Gon\c{c}alves' inequality states that $\|F\|_2^2$ is bounded below by $\abs{a_N}^2 (\prod_{n=1}^N \max\{1, \abs{\alpha_n}^2\}…

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…

Consider a trigonometric polynomial f of degree N, and associate to it the polynomial F in which each coefficient of f is replaced by its absolute value. F is called the majorant of f. We show that the L^3 norm of f can be larger than that…

This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…

The aim of this paper is to give an overview of some inequalities about $L^p$-norms ($p= 1$ or $p= 2$) of harmonic (periodic) and non-harmonic trigonometric polynomials. Among the material covered, we mention Ingham's Inequality about 2…

Although known lower bounds for the growth rate $\lambda$ of polyominoes, or Klarner's constant, are already close to the empirically estimated value $4.06$, almost no conceptual progress on upper bounds has occurred since the seminal work…

For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under…

The recurrence for the $k$-Fibonacci polynomials is usually iterated upwards to positive values of $n$ only. When the recurrence is iterated downwards to $n<0$, there are indices where the polynomials vanish identically. This fact does not…

We study polynomials that are orthogonal with respect to the modified Laguerre weight $z^{-n + \nu} e^{-Nz} (z-1)^{2b}$ in the limit where $n, N \to \infty$ with $N/n \to 1$ and $\nu$ is a fixed number in $\mathbb{R} \setminus…

A polynomial $p\in\mathbb{R}[z_1,\dots,z_n]$ is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the $z_1z_2\dots z_n$ monomial of a real stable…

The breakthrough paper of Croot, Lev, Pach \cite{CLP} on progression-free sets in $\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem…

The paper concerns the uniform polynomial approximation of a function $f$, continuous on the unit Euclidean sphere of ${\mathbb R}^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we…

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…