Seminar

Let k be a valued field, and let L be a finite extension of k(T_1,...T_n). If G is an abelian ordered group G containing |k^*|, every n-uple (g_1,..g_n) of elements of G gives rise to the so-called Gauss valuation  on k(T_1,...T_n), defined by the formula \sum a_I T^I \mapsto \max |a_I| g^I . I will explain how one can use a finiteness result by Hrushovski and Loeser concerning the space of stably dominated types (in the theory ACVF) on an algebraic curve to prove the following: there exists a finite subset E of L such that for every (g_1,...,g_n) as above, all extensions of the corresponding Gauss valuation to L are separated by elements of E. I will also say a few words about an application of this result (which was my main motivation for proving it) concerning subsets of Berkovich spaces that inherit a natural structure of piecewise-linear space.