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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.

### Most cited references4

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### Sharp estimates for the arithmetic Nullstellensatz

(2001)
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### Families of Dynamical Systems

(2007)
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### Portraits of preperiodic points for rational maps

(2014)
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