Andrei Bengus-Lasnier
Institute of Mathematics and Informatics - Bulgarian Academy of Sciences, Sofia, Bulgaria

My talk will be concerned with key polynomials and their potential ultrametric encoding. Key polynomials have been used to establish structure theorems for valued rings (e.g. generating sequences, valuative trees) in order to find strategies for proving local uniformization in positive characteristic.

My concern has been to build correspondences between these tools and more geometric objects, that would preserve the structure carried by these key polynomials. The simplest framework for key polynomials reduces to parametrizing extensions of valuations of a fixed valued field $(K,\nu)$ to a one variable polynomial ring $K[X]$. By passing to the algebraic closures $\overline{K}$ one can associate to any key polynomial a minimal pair $(a,\gamma)$ where $a\in\overline{K}$ and $\gamma\in \Gamma_\nu\otimes\mathbb{Q}$, $\Gamma$ being the value group of $\nu$. For a wide range of valuations (transcendental valuations) we have a perfect correspondence between key polynomials and associated minimal pairs.

I will present the basic theory of key polynomials as well as how to associate a minimal pair, thus giving us an ultrametric object to focus on: the ball with centre $a$ and radius $\gamma$.