Vol. 4, No. 4, 2011

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Preimages of quadratic dynamical systems

Benjamin Hutz, Trevor Hyde and Benjamin Krause

Vol. 4 (2011), No. 4, 343–363

For a quadratic polynomial with rational coefficients, we consider the problem of bounding the number of rational points that eventually land at a given constant after iteration, called preimages of the constant. It was shown by Faber, Hutz, Ingram, Jones, Manes, Tucker, and Zieve (2009) that the number of rational preimages is bounded as one varies the polynomial. Explicit bounds on the number of preimages of zero and 1 were addressed in subsequent articles. This article addresses explicit bounds on the number of preimages of any algebraic number for quadratic dynamical systems and provides insight into the geometric surfaces parameterizing such preimages.

quadratic dynamical systems, arithmetic geometry, preimage, rational points, uniform bound
Mathematical Subject Classification 2010
Primary: 37P05, 14G05
Secondary: 37F10
Received: 13 September 2010
Revised: 14 May 2011
Accepted: 15 May 2011
Published: 21 March 2012

Communicated by Bjorn Poonen
Benjamin Hutz
Department of Mathematics
CUNY Graduate Center
365 Fifth Ave
New York, NY 10016
United States
Trevor Hyde
Department of Mathematics and Computer Science
Amherst College
Amherst, MA 01002
United States
Benjamin Krause
Department of Mathematics
University of California
Box 951555
Los Angeles, CA 90095-1555
United States