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Uniform bounds on pre-images under quadratic dynamical systems. (English) Zbl 1222.11086
Summary: For any elements \(b,c\) of a number field \(K\), let \(G(b,c)\) denote the backwards orbit of \(b\) under the map \(f_c: \mathbb{C}\to\mathbb{C}\) given by \(f_c(x)=x^2+c\). We prove an upper bound on the number of elements of \(G(b,c)\) whose degree over \(K\) is at most some constant \(B\). This bound depends only on \(b\), \([K:Q]\), and \(B\), and is valid for all \(b\) outside an explicit finite set. We also show that, for any \(N>3\) and any \(b\) in \(K\) outside a finite set, there are only finitely many pairs of complex numbers \((y,c)\) for which \([K(y,c):K]<2^{(N-3)}\) and the value of the \(N\)-th iterate of \(f_c(x)\) at \(x=y\) is \(b\). Moreover, the bound \(2^{(N-3)}\) in this result is optimal.

MSC:
11G50 Heights
11G18 Arithmetic aspects of modular and Shimura varieties
14G05 Rational points
14G25 Global ground fields in algebraic geometry
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
Software:
Magma
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