Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

2020/02/17 by John R. Doyle, Bjorn Poonen · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals #Geometry and complex manifolds

paper · doi:10.1112/s0010437x20007022

Abstract

Fix d\geqslant 2 and a field k such that chark\nmid d . Assume that k contains the d th roots of 1 . Then the irreducible components of the curves over k parameterizing preperiodic points of polynomials of the form z^d+c are geometrically irreducible and have gonality tending to ∞ . This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of z^d+c . It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

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• Uniform bounds on periodic points of polynomials with good reduction
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