#### Vol. 4, No. 4, 2011

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### 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