Seminar
The p-adic valuation of the dynamics of polynomials, Dr. Luis A. Medina, University of Puerto Rico, Río Piedras, Thu, 11 December, 2025, 5:00 p.m. Student Center
Abstract
Let \(p\) be a prime. Suppose that \(f(t)=a_0+a_1t+\cdots+a_dt^d \in \mathbb{Z}_p[t].\) Define\begin{eqnarray*} x_1(c)&=&f(c)\\ x_n(c)&=&f(x_{n-1}(c)) \mathrm{\ for\ } n>1, \end{eqnarray*} where \(c \in \mathbb{Z}_p\). Consider the sequence \(\nu_p(x_n(c))_{n\geq1}\). As expected, the behavior of the sequence \(\nu_p(x_n(c))_{n\geq1}\) depends heavily on the polynomial \(f(t)\) and on the selected \(c\). In this work, we prove that, under certain conditions, the sequence \(\nu_p(x_n(c))_{n\geq1}\) is eventually constant. We also provide a full characterization of the behavior of \(\nu_p(x_n(c))_{n\geq1}\). Post is here.