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- Xander Faber, Benjamin Hutz, Patrick Ingram, Rafe Jones, Michelle Manes, Thomas J Tucker +1 other
- 2008

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)

- Robert L Benedetto, Dragos Ghioca, Benjamin Hutz, Pär Kurlberg, Thomas Scanlon, Thomas J Tucker +7 others
- 2011

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)

- Benjamin A Hutz, Sc M, Joseph Date, Silverman, Michael Date, Dan Rosen +4 others
- 2008

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)

- Benjamin Hutz
- 2009

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)

- Benjamin A Hutz
- 2008

This article examines dynamical systems on a class of K3 surfaces in P 2 ×P 2 with an infinite automorphism group. In particular, this article develops an algorithm to find Q-rational periodic points using information over F p for various primes p. This algorithm is then optimized to examine the growth of the average number of cycles versus p and to… (More)

The behavior under iteration of the critical points of a polynomial map plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us that fixing a particular critical point portrait and degree leads to only finitely many possible polynomials (up to… (More)

- Rafe Jones, Jordan Cahn, Jacob Spear, Robert Benedetto, Patrick Ingram, Alon Levy +12 others
- 2014

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)