We prove that if y = ∞ n=0 a(n)x n ∈ F q [[x]] is an algebraic power series of degree d, height h, and genus g, then the sequence a is generated by an automaton with at most q h+d+g−1 states, up to a vanishingly small error term. This is a significant improvement on previously known bounds. Our approach follows an idea of David Speyer to connect automata theory with algebraic geometry by representing the transitions in an automaton as twisted Cartier operators on the differentials of a curve.

Abstract. Let K be a number field or a function field of characteristic 0. If K is a number field, assume the abc-conjecture for K. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in K(x) that are not postcritically finite. For example, suppose K is a number field and f ∈ K[x] is not postcritically finite, and let K n be the field generated by the nth iterated preimages under f of β ∈ K. We show that for all large n, there is a prime of K that ramifies in K n and does not ramify in K m for any m < n.

Let K be the function field of a smooth, irreducible curve defined over Q. Let f ∈ K[x] be of the form f (x) = x q + c where q = p r , r ≥ 1, is a power of the prime number p, and let β ∈ K. For all n ∈ N ∪ {∞}, the Galois groups Gn(β) = Gal(K(f −n (β))/K(β)) embed into [Cq] n , the n-fold wreath product of the cyclic group Cq. We show that if f is not isotrivial, then [[Cq] ∞ : G∞(β)] < ∞ unless β is postcritical or periodic. We are also able to prove that if f1(x) = x q + c1 and f2(x) = x q + c2 are two such distinct polynomials, then the fields ∞ n=1 K(f −n 1 (β)) and ∞ n=1 K(f −n 2 (β)) are disjoint over a finite extension of K.

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 affine selfmap of P 1 (k) for k an algebraically closed field of positive characteristic. The Artin-Mazur Zeta FunctionLet X be a set and let f : X → X define a dynamical system. Let Per n (f ) = {x ∈ X : f n (x) = x}, where f n denotes the composition of f with itself n times. The Artin-Mazur zeta function of this dynamical system is the formal power series given byAssume that # Per n (f ) < ∞ for all n, as otherwise ζ is not defined. The power series ζ(f, X; t) has rational coefficients, and it is not hard to show that ζ(f, X; t) ∈ Z [[t]] by means of the product formulaThis zeta function was introduced by Artin and Mazur in [2], where it is studied for X a manifold and f a diffeomorphism. In this setting, only the isolated periodic points are counted. This will not be an important distinction for our purposes.This paper continues the study of the following question, introduced in [4] for polynomials, but just as easily phrased for rational functions.The purpose of this paper is to answer this question for rational maps that are dynamically affine. These are maps that, loosely speaking, come from endomorphisms of algebraic groups; a precise definition will be given in section 2. There are five families of dynamically affine maps in one dimension:

Given a finite endomorphism ϕ of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K(ϕ −∞ (α)) := n≥1 K(ϕ −n (α)) generated by the preimages of α under all iterates of ϕ. In particular when ϕ is post-critically finite, i.e. there exists a non-empty, Zariski-open W ⊆ X such that ϕ −1 (W ) ⊆ W and ϕ : W → X isétale, we prove that K(ϕ −∞ (α)) is ramified over only finitely many primes of K. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire [AHM05] in the case X = A 1 and Cullinan-Hajir, Jones-Manes [CH12, JM14] in the case X = P 1 . Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P 1 . The proof relies on Faltings' theorem and a local argument.