Abstract: Let $d$ be an integer greater than 1. In this talk, we will introduce a numerical criterion which bounds the degree of any algebraic integer in an interval of length less than 4 and we will discuss two applications in arithmetic dynamics. First, we show that there are only finitely many unicritical polynomials of the form $f_{c,d}(x)=x^d+c$ defined over the maximal totally real extension of rational numbers. Second, we classify quadratic unicritical polynomials $f_c(x)=x^2+c$ (where $c$ is a rational parameter) in which $f_c$ has only finitely many totally real (totally p-adic, respectively) preperiodic points. Our approach uses tools from arithmetic capacity theory and complex and p-adic dynamics.
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