Patrice Philippon

Groupes de Galois et nombres automatiques

Galois groups and automatic numbers : In the frame of Mahler’s method for algebraic independence, we show that the algebraic relations over Q linking the values of functions solutions of a system of functional equations come from the algebraic relations between the functions themselves, by specialisation. We deduce some new results on the linear independence of values of q-regular functions.

Quelques Aspects Diophantiens des VariéTés Toriques Projectives

Les varietes toriques jouent un role important au carrefour de l’algebre, la geometrie et la combinatoire. Elles constituent une classe de varietes suffisamment rigide pour que beaucoup des invariants s’explicitent en termes combinatoires, et en meme temps suffisamment riche pour permettre de tester et illustrer diverses conjectures et theories abstraites. Elle trouve application dans de nombreuses branches des mathematiques : geometrie algebrique bien sur, algebre commutative, combinatoire, calcul formel, geometries symplectique et kahlerienne, topologie et physique mathematique, voir par exemple [Ful93], [GKZ94], [Stu96], [Cox01], [Aud91], [Don02].

Height of varieties over finitely generated fields

We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply our previous work on toric varieties and extend our combinatorial formulae for the height to compute some arithmetic intersection numbers of non toric arithmetic varieties over the rational numbers.

Heights of Toric Varieties, Entropy and Integration over Polytopes

We present a dictionary between arithmetic geometry of toric varieties and convex analysis. This correspondence allows for effective computations of arithmetic invariants of these varieties. In particular, combined with a closed formula for the integration of a class of functions over polytopes, it gives a number of new values for the height (arithmetic analog of the degree) of toric varieties, with respect to interesting metrics arising from polytopes. In some cases these heights are interpreted as the average entropy of a family of random processes.