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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
##### Keywords:
polynomial dynamics; function fields; Riccati equations
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