### "On analytic functions in an ordered field with a Krull valuation"

#### Dr. Héctor Moreno Barrera, Departamento de Matemáticas ULS

"Ultrametric Calculus over valued fields of rank one is a well developed theory (p-adic numbers field, Levi Civita field). In the case of fields with valuations of infinite rank, R. Hobeika (Ph.D Thesis, University of Paris VI, 1976) studied power series and Laurent series on Krull valued fields F, and showed that the domain of convergence of a power series is either K or reduced to a point. Therefore the classical definition of an analytic function on an open subset only gives entire functions on K. Y. Perrin extended the notion of analytic functions on an open subset, defining it as a limit of rational functions. However the fields she considers are always algebraically closed.

At this talk, we will present results concerning analytic functions defined on a non-archimedean ordered field K, using the fact the order of K induces a Krull valuation, in addition that they induce a common topology. We also give counterexamples to some theorems for the real or complex case".