Close p-adic Roots of Integer Polynomials

In this thesis I will study the question of how close roots of a fixed degree integer polynomial can be in terms of its height. This question has been studied extensively in the case of real and complex roots but there is much less known in the case of $p$-adic roots and this is what will be studied in this work. More specifically we will investigate the largest real number $K$ such that for any $\kappa < K$ it is possible to find infinitely many integer polynomials of fixed degree and bounded height such that
$$|\alpha_1-\alpha_2|_p\leq H(P)^{-\kappa}$$
holds for some roots $\alpha_1 \neq \alpha_2\in \overline{\mathbb{Q}_p}$ of $P$. This question for the case of real and complex roots was first discussed by Mahler in 1964 as he proved that $K \leq n+1$ for polynomials of degree $n$.

I will also study the quantitative version of this problem by counting the number of polynomials with close $p$-adic roots. We will also explore the related problem of bounding the discriminant of polynomials as we consider the separation of all roots of a polynomial. To this end we will establish a counting result for the number of polynomials of fixed degree and bounded height with $p$-adically small discriminant.

The method that I use relies on the quantitative non-divergence of Kleinbock and Tomanov and its use in the investigation of the distribution of close $p$-adic roots to find the infimum $K$. This work follows and develops similar methods for the real and complex case by Beresnevich, Bernik and G\"otze.