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New families satisfying the dynamical uniform boundedness principle over function fields

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Abstract

We extend a technique, originally due to the first author and Poonen, for proving cases of the Strong Uniform Boundedness Principle (SUBP) in algebraic dynamics over function fields of positive characteristic. The original method applied to unicritical polynomials for which the characteristic does not divide the degree. We show that many new 1-parameter families of polynomials satisfy the SUBP, including the family of all quadratic polynomials in even characteristic. We also give a new family of non-polynomial, non-Lattès rational functions that satisfies the SUBP.

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Notes

  1. For the SUBP to hold in this setting, one must generally exclude those parameters c lying in the constant subfield.

  2. The gonality of an irreducible k-curve C is the minimum degree of a nonconstant k-morphism \(C \rightarrow {\mathbb {P}}^1\).

  3. To define the reduction modulo \({\mathfrak {p}}\), one should normalize f so that the minimum \({\mathfrak {p}}\)-adic valuation of the coefficients is 0.

  4. The content of this statement comes from the fact that a rational function f may admit infinitely many nontrivial twists: rational functions conjugate to f over \(\overline{K}\), but not over K.

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Acknowledgements

The first author was partially supported by NSF grant DMS-2112697. We thank Laura DeMarco and Nicole Looper for their feedback on this work, and we thank the anonymous referee for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, USA

    John R. Doyle

  2. Institute for Defense Analyses, Center for Computing Sciences, Bowie, MD, USA

    Xander Faber

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  1. John R. Doyle
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  2. Xander Faber
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Correspondence to John R. Doyle.

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Doyle, J.R., Faber, X. New families satisfying the dynamical uniform boundedness principle over function fields. Math. Ann. 388, 985–1000 (2024). https://doi.org/10.1007/s00208-022-02536-z

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