Benjamin Hutz

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For any elements a, c of a number field K, let Γ(a, c) denote the backwards orbit of a under the map fc : C → C given by fc(x) = x 2 + c. We prove an upper bound on the number of elements of Γ(a, c) whose degree over K is at most some constant B. This bound depends only on a, [K : Q], and B, and is valid for all a outside an explicit finite set. We also(More)
Let K be a number field, let ϕ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K. We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present(More)
iv Acknowledgements I would first like to thank my advisor Joseph Silverman, whose knowledge and keen insight were invaluable to the completion of this work. Our weekly discussions never failed to provide more ideas to ponder and a clearer understanding of the current problems. His careful reading also kept many errors out of this work that I would have(More)
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article " Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems, " by the present authors and five others, it was shown that the number of rational(More)
Research Interests • Galois theory and irreducibility of polynomials (particularly involving iterated morphisms) • Discrete dynamics (particularly iteration of rational functions over global and finite fields) • Automorphism groups of rooted trees and iterated monodromy groups • Elliptic curves and torsion fields • Recurrence sequences Perfect powers in(More)