Arithmetic Dynamical Questions with Local Rationality Conditions

**Note special time for this talk**

Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo

In this talk, we first introduce a numerical criterion which bounds the degree of any algebraic integer in short intervals. Then we use the numerical tool to classify all the unicritical polynomials defined over the maximal totally real extension of the field of rational numbers. We also classify all the quadratic unicritical polynomials defined over the field of rational numbers in which they have only finitely many totally real preperiodic points. Using the numerical tool, we are able to explicitly compute totally real preperiodic points of some quadratic unicritical polynomials. Our approach uses tools from complex and p-adic potential theory, particularly, the Fekete-Szego theorem. This is based on joint work with Clay Petsche.

MC 5501