Related papers: Misiurewicz polynomials for rational maps with non…

We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$.…

We present a more general proof that cyclotomic polynomials are irreducible over Q and other number fields that meet certain conditions. The proof provides a new perspective that ties together well-known results, as well as some new…

We prove that the closure (for the Zariski topology) of the set of polynomial automorphisms of the complex affine plane whose polydegree is (cd-1,b,a) contains all triangular automorphisms of degree cd+a, where a,b >1 and c>0 are integers…

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…

We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov…

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational…

We construct explicit non-isotrivial families of polynomials over $\mathbb{Q}$ satisfying uniform boundedness for their rational preperiodic points.

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form…

We continue the analysis of the Modular Isomorphism Problem for $2$-generated $p$-groups with cyclic derived subgroup, $p>2$, started in [D. Garc\'ia-Lucas, \'A. del R\'io, and M. Stanojkovski. On group invariants determined by modular…

It is conjectured that a rational map whose coefficients are algebraic over $\Q_p$ has no wandering components of the Fatou set. R. Benedetto has shown that any counter example to this conjecture must have a wild recurrent critical point.…

We present a natural extension of the notion of nondegenerate rational maps (quadrirational maps) to arbitrary dimensions. We refer to these maps as $2^n-$rational maps. In this note we construct a rich family of $2^n-$rational maps. These…

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the…

We give a lower bound for the degree of an irreducible factor of a given polynomial. This improves and generalizes the results obtained in [4, On the irreducible factors of a polynomial, Proc. Amer. Math. Soc., 148 (2020] 1429 -- 1437].

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes…

We classify all solutions (p,q) to the equation p(u)q(u)=p(u+b)q(u+a) where p and q are complex polynomials in one indeterminate u, and a and b are fixed but arbitrary complex numbers. This equation is a special case of a system of…

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…

Let $I(X,R)$ be the incidence algebra of the preordered set $X$ over the ring $R$. In the case of a finite connected partially ordered set $X$, we prove that the subgroup of inner multiplicative automorphisms is a direct factor of the group…

For a classical group $G$ of type $\mathsf D_n$ over a field $k$ of characteristic different from $2$, we show the existence of a finitely generated regular extension $R$ of $k$ such that $G$ admits outer automorphisms over $R$. Using this…

A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals…