University of Connecticut

Algebra Seminar

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Title: Prime divisors in orbits and Galois groups of iterates
Speaker: Wade Hindes (City University of New York)
Time: Wednesday, November 15, 2017 at 11:15 am
Place: MONT 313Abstract: Given a global field $K$ and a polynomial $\phi \in K[x]$, 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 $K$ is a finite extension of $\mathbb{F}_p(t)$, 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 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.​


Organizer: Liang Xiao