Related papers: Preperiodic points of polynomial dynamical systems…

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…

Let g >= 1 and let Q be a monic, squarefree polynomial of degree 2g + 1 in Z[x]. For an odd prime p not dividing the discriminant of Q, let Z_p(T) denote the zeta function of the hyperelliptic curve of genus g over the finite field F_p…

We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive…

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$…

For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period…

In this paper, we show that each expanding Thurston map $f : S^2\rightarrow S^2$ has $1+ deg f$ fixed points, counted with appropriate weight, where $ deg f$ denotes the topological degree of the map $f$. We then prove the equidistribution…

We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor.In particular, we show that the module of syzygies of polynomials…

For X a finite subset of the circle and for 0 < r <= 1 fixed, consider the function f_r : X -> X which maps each point to the clockwise furthest element of X within angular distance less than 2 pi r. We study the discrete dynamical system…

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors.…

We establish estimates for exact upper bounds of deviations of partial Fourier sums $S_{n-1}(f)$ on classes $W^r_{\beta,1}, r>2, \beta\in\mathbb{R},$ of $2\pi$-periodic functions whose $(r,\beta)$-derivatives in the Weyl--Nagy sense belong…

For positive integers $a$ and $b$, we let $[U_n]$ be the Lucas sequence of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{ for $n\ge 2$},\] and let $\pi(m):=\pi_{(a,b)}(m)$ be the…

An algorithm is presented here, for discovering Hopf-Bifurcation varieties of polynomial dynamical systems. It is based on the expression of specific polynomials, as sums of products of first degree polynomials, with parametrical…

Uniform asymptotic formulae for arrays of complex numbers of the form $(f_{r,s})$, with $r$ and $s$ nonnegative integers, are provided as $r$ and $s$ converge to infinity at a comparable rate. Our analysis is restricted to the case in which…

Let K be a non-archimedean field, and let f in K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of f and their preimages, that determines…

Let $f_1(x),\ldots,f_n(x)$ be some polynomials. The upper bound on the number of $x\in\mathbb F_p$ such that $f_1(x),\ldots,f_n(x)$ are roots of unit of order $t$ is obtained. This bound generalize the bound of the paper \cite{V-S} to the…

Let $f_{c,d}(x)=x^d+c\in \mathbb{C}[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\bar{\mathbb{Q}}$. Suppose that the Misiurewicz…

In the space $\mathbb{C}$ of the parameters $\lambda$ of the unicritical polynomials family $f(\lambda,z)=f_\lambda(z)=z^d+\lambda$ of degree $d>1$, we establish a quantitative equidistribution result towards the bifurcation current (indeed…

Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic…

For any positive integers $n\geq 3, r\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our…

Let $F$ be a number field. Given a quadratic polynomial $f_c(z) = z^2 + c \in F[z]$, we can construct a directed graph $Preper(f_c, F)$ (also called a portrait), whose vertices are $F$-rational preperiodic points for $f_c$, with an edge…