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Riccati equations and polynomial dynamics over function fields. (English) Zbl 07174529
Authors’ abstract: Given a function field \(K\) and \(\phi \in K[x]\), we study two finiteness questions related to iteration of \(\phi\): whether all but finitely many terms of an orbit of \(\phi\) must possess a primitive prime divisor, and whether the Galois groups of iterates of \(\phi\) must have finite index in their natural overgroup \(\mathrm{Aut}(T_d)\), where \(T_d\) is the infinite tree of iterated preimages of \(0\) under \(\phi\). We focus particularly on the case where \(K\) has characteristic \(p\), where far less is known. We resolve the first question in the affirmative for a large (in particular, Zariski-dense) subset of the space of degree-\(d\) polynomials. The main step in the proof is to rule out certain algebraic relations among points in backwards orbits; these relations are given by a type of first-order differential equation called a Riccati equation. We then apply our result on primitive prime divisors and adapt a method of N. Looper [Bull. Lond. Math. Soc. 51, No. 2, 278–292 (2019; Zbl 07094881)] to produce new families of polynomials in every characteristic for which the second question has an affirmative answer. We also prove that almost all quadratic polynomials over \(\mathbb{Q}(t)\) have iterates whose Galois group is all of \(\mathrm{Aut}(T_d)\).

MSC:
11R32 Galois theory
37P15 Dynamical systems over global ground fields
14G05 Rational points
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