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- W2762505787 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 $text{Aut}(T_d)$, 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 under relatively weak hypotheses; interestingly, the main step in our proof is to rule out Riccati differential equations in backwards orbits. We then apply our result on primitive prime divisors and adapt a method of Looper to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials. We also prove that almost all quadratic polynomials over $mathbb{Q}(t)$ have iterates whose Galois group is all of $text{Aut}(T_d)$." @default.
- W2762505787 created "2017-10-20" @default.
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- W2762505787 date "2017-10-12" @default.
- W2762505787 modified "2023-09-27" @default.
- W2762505787 title "Riccati equations and polynomial dynamics over function fields" @default.
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