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Hindes, Wade

doi:10.1016/j.jnt.2014.09.033

Cited by 14 publications

(23 citation statements)

References 13 publications

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“…It is tempting to think that the discriminant trick we used to prove G Kpµ d q pφq -W pdq for φpxq " x d`t works for all φpxq " x d`f satisfying f R Kpµ d q d . However, surjectivity already fails for d " 2: when φpxq " x 2´p t 2`1 q, we show in [12] that rW p2q : G K pφqs " 2, even though´pt 2`1 q is never a square in characteristic zero; this example is essentially due to Stoll [45]. Likewise, this discriminant trick does not work in prime characteristic: when d " 5, p " 43 and φpxq " x 5`t , the discriminant of φ 6 p0q is zero in F p rts.…”

Section: Dynamical Galois Groups and The Kodaira-spencer Mapmentioning

confidence: 72%

“…As for characteristic zero function fields, we can make the index bounds explicit and uniform when K " kptq is a rational function field; compare to uniform bounds in the quadratic case [12]. Here we use work of Schmidt [34] and Mason [25] on Thue Equations over function fields; conveniently, we need not worry about isotriviality, since it does not affect the height bounds in this setting.…”

Section: Dynamical Galois Groups and The Kodaira-spencer Mapmentioning

confidence: 99%

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Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map

Hindes

2018

Trans. Amer. Math. Soc.

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Let K be a global function field and let φ P Krxs. For all wandering basepoints b P K, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set Zpφ, bq that depends only on φ, the poles of the b, and K. Moreover, when we order b P O K,S by height, we show that Zpφ, bq is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of φpxq " x d`f is a finite index subgroup of an iterated wreath product of cyclic groups. Finally, we establish similar results on Zsigmondy sets when K is the field of rational numbers or K is a quadratic imaginary field subject to an added stipulation: either zero has finite orbit under iteration of φ or the Vojta conjecture for algebraic points on curves holds.

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Section: Dynamical Galois Groups and The Kodaira-spencer Mapmentioning

confidence: 72%

Section: Dynamical Galois Groups and The Kodaira-spencer Mapmentioning

confidence: 99%

Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map

Hindes

2018

Trans. Amer. Math. Soc.

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“…To prove Theorem 1, we build on our techniques from the characteristic zero setting [14]. There, among other things, we prove the uniform bound (4) #Zpφ, γ, 2q ď 17…”

Section: Primitive Prime Divisors and Superelliptic Curvesmentioning

confidence: 99%

Prime divisors in polynomial orbits over function fields

Hindes

2016

Bull. London Math. Soc.

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Given a polynomial ϕ over a global function field K/double-struckFq(t) and a wandering base point b∈K, we give a geometric condition on ϕ, ensuring the existence of primitive prime divisors for almost all points in the orbit MJX-tex-caligraphicscriptOϕ(b):={ϕn(b)}n⩾0. As an application, we prove that the Galois groups (over K) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of MJX-tex-caligraphicscriptOϕ(b).

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“…Acknowledgements. I would like to thank Laura DeMarco for helpful discussions regarding this article, and Wade Hindes for introducing me to questions of Galois uniformity through his thesis [5].…”

Section: Introductionmentioning

confidence: 99%