Title : Rational functions and piecewise-linear maps in non-archimedean geometry

Abstract : Let X be an irreducible algebraic variety over a non-archimedean field k. Vladimir Berkovich has attached

to it an analytic space X^{an} with very good topological properties (it is locally compact and locally arcwise connected), 

and which contains plenty of "natural" piecewise-linear spaces called skeleta. More precisely, if S is a skeleton, 

its piecewise-linear structure is natural in the sense that it is "induced by rational functions on X":
- if f is a non-zero rational function on X, log |f| is well-defined on S (since the latter consists of Zariski-dense points), and is PL (piecewise linear);
- there exist finitely many non-zero rational functions f_1,..f_m on X such that the log lf_i| induce a PL-isomorphism between S and a finite union 

of convex polytopes in R^m.

In this talk, I will present a joint work with Ehud Hrushovski, Francois Loeser and Jinhe He, in which we prove by model-theoretic methods, 

under the assumption that the ground field k is algebraically closed, a finiteness result for the set of PL functions on a given skeleton S 

of the form log |f| with f rational and non-zero. More precisely, we show that there exist non-zero rational functions g_1,..., g_r on X such 

that the functions on S of the form log |f| are exactly those that can be obtained using only the +,-,min and max operations from 

the log |g_i| and the constant functions with values in log |k^*|.