For any real polynomial $p(x)$ of even degree k, Shapiro [‘Problems around polynomials: the good, the bad and the ugly$\ldots $’, Arnold Math. J. 1(1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$, where $C(x)$ is a polynomial with $C(w)\ne 0$, and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial $f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ we show that the Mahler measure of the iterates $f^n$ grows geometrically fast with the degree $d^n,$ and find the exact base of that exponential growth. This base is expressed via an integral of $\log ^+|z|$ with respect to the invariant measure of the Julia set for the polynomial $f.$ Moreover, we give sharp estimates for such an integral when the Julia set is connected.

Classical results about peaking from complex interpolation theory are extended to polynomials on a closed disk, and on the complement of its interior. New results are obtained concerning interpolation by univalent polynomials on a Jordan domain whose boundary satisfies certain smoothness conditions.

The classical Gauss–Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows.

We show that if the coefficients of a monic polynomial $p(z)$ are in the sector {tei𝜓 : 𝜓∈ [0, 𝜙], t⩾0}, for some $\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x1D70B})$, and the zeros are not in its interior, then the critical points of $p(z)$ are also not in the interior of that sector.

In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result.

Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e., has only real roots), then $p+s{{p}^{'}}$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form ${{p}_{a}}(z,s)\,\,:=\,\,\,p\,(z)+\,\sum\nolimits_{k=1}^{d}{{{a}_{k}}{{s}^{k}}{{p}^{(k)}}(z)}$. We give a full characterization of those $a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$ for which ${{p}_{a}}(z,s)$ is a pencil of hyperbolic polynomials. We also give a full characterization of those $a=({{a}_{1}},...,{{a}_{d}})\,\in \,{{\mathbb{R}}^{d}}$ for which the associated families $ $ admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.

A Littlewood polynomial is a polynomial in $\mathbb{C}\left[ z \right]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small ${{L}^{q}}$ normon the complex unit circle. We consider the Fekete polynomials

$${{f}_{p}}(z)=\sum\limits_{j=1}^{p-1}{(j|p){{z}^{j}}},$$

where $p$ is an odd prime and $(.|p)$ is the Legendre symbol (so that ${{z}^{-1}}{{f}_{p}}(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of ${{L}^{q}}$ and ${{L}^{2}}$ norm of ${{f}_{p}}$ when $q$ is an even positive integer and $p\to \infty $. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the ${{L}^{4}}$ norm of these polynomials.

Let ${{A}_{n}}\,=\,\{{{a}_{0}}+{{a}_{1}}z+\,.\,.\,.\,+\,{{a}_{n-1}}{{z}^{n-1}}\,:\,{{a}_{j}}\,\in \,\{0,1\}\}$ , whose elements are called zero-one polynomials and correspond naturally to the ${{2}^{n}}$ subsets of $\left[ n \right]\,:=\,\{0,\,1,\,.\,.\,.\,,\,n-1\}$. We also let ${{A}_{n,m\,}}\,=\,\{\alpha \left( z \right)\,\in \,{{A}_{n}}\,:\,\alpha \left( 1 \right)\,=\,m\}$ , whose elements correspond to the $\left( _{m}^{n} \right)$ subsets of $\left[ n \right]$ of size $m$, and let ${{B}_{n}}\,=\,{{A}_{n+1}}\,\backslash \,{{A}_{n}}$ , whose elements are the zero-one polynomials of degree exactly $n$.

Many researchers have studied norms of polynomials with restricted coefficients. Using $\|\alpha {{\|}_{p}}$ to denote the usual ${{L}_{p}}$ norm of $\alpha$ on the unit circle, one easily sees that $\alpha \left( z \right)\,=\,{{a}_{0}}+{{a}_{1}}z+.\,.\,.+{{a}_{N}}{{z}^{N}}\,\in \,\mathbb{R}\left[ z \right]$ satisfies $\|\alpha \|_{2}^{2}\,=\,{{c}_{0}}$ and $\|\alpha \|_{4}^{4}\,=\,c_{0}^{2}\,+\,2\left( c_{1}^{2}\,+\,.\,.\,.\,+\,c_{N}^{2} \right)$ , where ${{c}_{k}}\,:=\,\sum _{j=0}^{N-k}\,{{a}_{j}}{{a}_{j+k}}\,\text{for}\,\text{0}\le \,\text{k}\,\le N$.

If $\alpha \left( z \right)\,\in \,{{A}_{n,m}}$ , say $\alpha \left( z \right)\,=\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{1}}}}\,+\,.\,.\,.\,+\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{m}}}}$ where ${{\beta }_{1}}\,<\,.\,.\,.\,<\,{{\beta }_{m}}$ , then ${{c}_{k}}$ is the number of times $k$ appears as a difference ${{\beta }_{i}}\,-\,{{\beta }_{j}}$ . The condition that $\alpha \,\in \,{{A}_{n,m}}$ satisfies ${{c}_{k}}\,\in \,\{0,1\}$ for $1\,\le \,k\,\le \,n\,-\,1$ is thus equivalent to the condition that $\{{{\beta }_{1}},\,.\,.\,.\,,\,{{\beta }_{m}}\}$ is a Sidon set (meaning all differences of pairs of elements are distinct).

In this paper, we find the average of $\left\| \alpha \right\|_{4}^{4}$ over $\alpha \,\in \,{{A}_{n}}$, $\alpha \,\in \,{{B}_{n}}$ and $\alpha \,\in \,{{A}_{n,m}}$. We further show that our expression for the average of $\left\| \alpha \right\|_{4}^{4}$ over ${{A}_{n,m}}$ yields a new proof of the known result: if $m\,=\,o\left( {{n}^{1/4}} \right)$ and $B\left( n,\,m \right)$ denotes the number of Sidon sets of size $m$ in $\left[ n \right]$, then almost all subsets of $\left[ n \right]$ of size $m$ are Sidon, in the sense that ${{\lim }_{n\to \infty }}\,B\left( n,\,m \right)/\left( _{m}^{n} \right)\,=\,1$.

A well-known result, due to Ostrowski, states that if ${{\left\| P-Q \right\|}_{2}}\,<\,\varepsilon $, then the roots $({{x}_{j}})$ of $P$ and $({{y}_{j}})$ of $Q$ satisfy $\left| {{x}_{j}}\,-\,{{y}_{j}} \right|\,\le \,Cn{{\varepsilon }^{1/n}}$, where $n$ is the degree of $P$ and $Q$. Though there are cases where this estimate is sharp, it can still be made more precise in general, in two ways: first by using Bombieri’s norm instead of the classical ${{l}_{1}}$ or ${{l}_{2}}$ norms, and second by taking into account the multiplicity of each root. For instance, if $x$ is a simple root of $P$, we show that $\left| x\,-\,y \right|\,<\,C\varepsilon $ instead of ${{\varepsilon }^{1/n}}$. The proof uses the properties of Bombieri’s scalar product andWalsh Contraction Principle.

For a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.

In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show that

and for a root of unity α that

We study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

This paper addresses a number of questions concerning the size of factors of polynomials. Let p be a monic algebraic polynomial of degree n and suppose q1q2 … qi is a monic factor of p of degree m. Then we can, in many cases, exactly determine

Here ‖ . ‖ is the supremum norm either on [—1, 1] or on {|z| ≤ 1}. We do this by showing that, in the interval case, for each m and n, the n-th Chebyshev polynomial is extremal. This extends work of Gel'fond, Mahler, Granville, Boyd and others. A number of variants of this problem are also considered.

Let be the class of all polynomials/? of degree at most n such that In view of the example zn it follows from Bernstein's inequality for polynomials that at each point z0 of the unitcirle. It was shown by A. Giroux and Q. I. Rahman [2] that if denotes the subclass of polynomials in which vanish at 1, then

where c1 and c2 are constants not depending on n. Here we find the exact value of which has some special significance and also at certain other points of the unit circle.