Preimages of quadratic dynamical systems

Abstract

(Communicated by Bjorn Poonen)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.

Rational preimages in families of dynamical systems

“…Along the same lines, for any number field K and a ∈ K, Hutz, Hyde, and Krause [11] have computed explicit sharp upper bounds κ(a, K) ∈ {4, 6, 8, 10} such that |Preim(φ, a, K)| ≤ κ(a, K) for all but finitely many φ ∈ F (K).…”

“…Let C = deg(b 2 − 4c + 4a). Then from(11), ord P y 1 = C 2 ord P t. So ord P y n−1 ≥ min e, C 2 n−1 ord P t.…”