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Answering a question asked by Hsia and Tucker in their paper on the finiteness of greatest common divisors of iterates of polynomials, we prove that if $f, g \in \mathbb{C}(X)$ are compositionally independent rational functions and $c \in…

We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity…

Let $q = p^s$ be a power of a prime number $p$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over $\mathbb{F}_q$ where both $r…

In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an…

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative…

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must…

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois…

Let f(x) be a polynomial of degree at least 5 with complex coefficients and without repeated roots. Let p be an odd prime. Suppose that all the coefficients of f(x) lie in a subfield K such that: 1) K contains a primitive p-th root of…

For every nonconstant polynomial $f\in\mathbb Q[x]$, let $\Phi_{4,f}$ denote the fourth dynatomic polynomial of $f$. We determine here the structure of the Galois group and the degrees of the irreducible factors of $\Phi_{4,f}$ for every…

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's…

We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow…

Let K be any field and G be a finite group. Noether's problem asks whether the fixed field is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p^n containing a cyclic subgroup of index p…

Let $\phi$ be a rank $r$ Drinfeld $\BF_q[T]$-module determined by $\phi_T(X) = TX+g_1X^q+...+g_{r-1}X^{q^{r-1}}+X^{q^r}$, where $g_1,...,g_{r-1}$ are algebraically independent over $\BF_q(T)$. Let $N\in\BF_q[T]$ be a polynomial, and…

Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in \mathbb{P}^1(K)$, we study the extension $K(\phi^{-\infty}(b))$ generated by the union over $n$ of all solutions to $\phi^n(x) = b$, where $\phi^n$ is the $n$th…

We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage…

Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the…

We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems…

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field:…

Given a field $K$, a polynomial $f \in K[x]$, and a suitable element $t \in K$, the set of preimages of $t$ under the iterates $f^{\circ n}$ carries a natural structure of a $d$-ary tree. We study conditions under which the absolute Galois…

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),…