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Automatic Sequences and Curves over Finite Fields
We prove that if $y=\sum_{n=0}^\infty{\bf a}(n)x^n\in\mathbb{F}_q[[x]]$ is an algebraic power series of degree $d$, height $h$, and genus $g$, then the sequence ${\bf a}$ is generated by an automatonExpand
On the number of distinct functional graphs of affine-linear transformations over finite fields
Abstract We study the number of non-isomorphic functional graphs of affine-linear transformations from ( F q ) n to itself, and we prove upper and lower bounds on this quantity as n→∞. As a corollaryExpand
Given a finite endomorphism $\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1}Expand
The Artin-Mazur Zeta Function of a Dynamically Affine Rational Map in Positive Characteristic
A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affineExpand
Dynamically distinguishing polynomials
A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: For any prime p, reduce its coefficients mod p and consider its action on the fieldExpand
The Generalized Nagell–Ljunggren Problem: Powers with Repetitive Representations
We consider a natural generalization of the Nagell–Ljunggren equation to the case where the qth power of an integer y, for q ⩾ 2, has a base-b representation that consists of a length-ℓ block of digits repeated n times. Expand
Finite index theorems for iterated Galois groups of cubic polynomials
Let K be a number field or a function field. Let $$f\in K(x)$$f∈K(x) be a rational function of degree $$d\ge 2$$d≥2, and let $$\beta \in {\mathbb {P}}^1(\overline{K})$$β∈P1(K¯). For all $$n\inExpand
Finite index theorems for iterated Galois groups of unicritical polynomials.
Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the primeExpand
Zeta Functions of Polynomial Dynamics on the Algebraic Closure of a Finite Field
We study the rationality of the Artin-Mazur zeta function of a dynamical system defined by a polynomial map on the algebraic closure of the finite field F_p. The zeta functions of the maps f(x)=x^mExpand