Dynamical Galois groups of trinomials and Odoni's conjecture

Nicole R. Looper

Dynamical Galois groups of trinomials and Odoni's conjecture

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arxiv 1609.03398 v2 pith:YDH7O5TV submitted 2016-09-12 math.NT

Nicole R. Looper This is my paper · ORCID

classification math.NT

keywords conjectureprimedegreesgaloisodonipolynomialprovearboreal

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abstract

We prove Odoni's conjecture in all prime degrees; namely, we prove that for every positive prime $p$, there exists a degree $p$ polynomial $\varphi\in\mathbb{Z}[x]$ with surjective arboreal Galois representation. We also show that Vojta's conjecture implies the existence of such a polynomial in many degrees $d$ which are not prime.

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A note on Misiurewicz polynomials
math.NT 2019-08 conditional novelty 7.0 of 10

For prime d, the Misiurewicz polynomial G_{d,m,n} has no more rational irreducible factors than its reduction modulo d, which yields new irreducibility families and the first bounds for periods above three.

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