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      Orbits of Polynomial Dynamical Systems Modulo Primes

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          Abstract

          We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over \(\mathbb C\). Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly improve a result of Silverman~(2008) and recover a result of Akbary and Ghioca~(2009) as special extreme cases of our estimates.

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          Sharp estimates for the arithmetic Nullstellensatz

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            Families of Dynamical Systems

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              Portraits of preperiodic points for rational maps

              Let \(K\) be a function field over an algebraically closed field \(k\) of characteristic \(0\), let \(\varphi\in K(z)\) be a rational function of degree at least equal to \(2\) for which there is no point at which \(\varphi\) is totally ramified, and let \(\alpha\in K\). We show that for all but finitely many pairs \((m,n)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}\) there exists a place \(\mathfrak{p}\) of \(K\) such that the point \(\alpha\) has preperiod \(m\) and minimum period \(n\) under the action of \(\varphi\). This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also when \(\varphi\) has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple \((c_1,\dots , c_{d-1})\in k^{n-1}\) and for almost all pairs \((m_i,n_i)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}\) for \(i=1,\dots, d-1\), there exists a polynomial \(f\in k[z]\) of degree \(d\) in normal form such that for each \(i=1,\dots, d-1\), the point \(c_i\) has preperiod \(m_i\) and minimum period \(n_i\) under the action of \(f\).
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                Author and article information

                Journal
                2017-02-07
                Article
                1702.01976

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.NT

                Number theory

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