#### 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
##### Abstract

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.

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

Communicated by Bjorn Poonen
##### Authors
 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