On periods modulo \(p\) in arithmetic dynamics
This work deals with the problem of simultaneous pre-periodicity in parameterised families of algebraic dynamical systems. The author considers two parametrised families of polynomial maps over an algebraic number field, and investigates the size modulo \(p\) of the orbit of their critical point. The polynomials have a specific form: \(z^d+t\) and \(z^d+h(t)\), where \(d\geqslant 2\), \(t\) is the parameter, and \(h\) is an arbitrary polynomial -- as long as it is not linear and of finite order. One then considers the local dynamics in the (algebraic closure of the) residue field of a prime ideal of good reduction, allowing the parameter to range in that field. The result is a lower bound for the size of the critical orbit of at least one of the two maps. The author shows that there are two constants \(c_1\) and \(c_2\) depending only on \(d\) and \(h\), such that for almost all prime ideals (in the sense of density) and all but \(c_1\) parameter values, the critical orbit of at least one of the two maps consists of at least \(c_2\log N\) points, where \(N\) is the absolute norm of the ideal. This result is related to earlier work of \textit{J. H. Silverman} [New York J. Math. 14, 601--616 (2008; Zbl 1153.11028)], of \textit{A. Akbary} and \textit{D. Ghioca} [J. Number Theory 129, No. 11, 2831--2842 (2009; Zbl 1254.37065)], and of \textit{D. Ghioca}, \textit{H. Krieger} and \textit{K. Nguyen} [``A case of the dynamical Andre-Oort conjecture'', Preprint, \url{arXiv:1404.6011}].
scientific article; zbMATH DE number 6424577