Related papers: Maximal and linearly inextensible polynomials

Cyclotomic polynomials play fundamental roles in number theory, combinatorics, algebra and their applications. Hence their properties have been extensively investigated. In this paper, we study the maximum gap $g$ (maximum of the…

Given a lower envelope in the form of an arbitrary sequence $u$, let $LSP(u, d)$ denote the maximum length of any subsequence of $u$ that can be realized as the lower envelope of a set of polynomials of degree at most $d$. Let $sp(m, d)$…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the…

In this paper, we examine how far a polynomial in $\mathbb{F}_2[x]$ can be from a squarefree polynomial. For any $\epsilon>0$, we prove that for any polynomial $f(x)\in\mathbb{F}_2[x]$ with degree $n$, there exists a squarefree polynomial…

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of…

Let $f$ be a non-zero polynomial with complex coefficients and define $M_n(f)=\int_0^1f(x)^n\,dx$. We use ideas of Duistermaat and van der Kallen to prove $\limsup_{n\rightarrow\infty}|M_n(f)|^{1/n}>0$. In particular, $M_n(f)\ne 0$ for…

Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $\epsilon=\epsilon(\mathbb{P})>0$ such that, for any compact set $E \subset…

We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of…

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its…

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…

Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random…

In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection…

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying…

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting…

We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$, prove its completeness for polynomials of degrees $\leq 4$, calculate all 71 possible values of these invariants for the case of…

Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q),…