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• W2964332149 startingPage "1555" @default.
• W2964332149 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 $operatorname {Aut}(T_d)$, 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 Looper 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 $operatorname {Aut}(T_d)$." @default.
• W2964332149 created "2019-07-30" @default.
• W2964332149 creator A5001570714 @default.
• W2964332149 creator A5028002831 @default.
• W2964332149 date "2019-12-10" @default.
• W2964332149 modified "2023-10-02" @default.
• W2964332149 title "Riccati equations and polynomial dynamics over function fields" @default.
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• W2964332149 doi "https://doi.org/10.1090/tran/7855" @default.
• W2964332149 hasPublicationYear "2019" @default.
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