| Résume | Transseries are formal series, involving exponentials and logarithms, which provide a natural model for the model theory of Hardy fields, i.e. ordered fields of real-valued differentiable germs. Transseries are not closed under many functional equations, whose solutions still behave like germs in Hardy fields. Hyperexponential functions, which grow faster or than any finite iteration of the exponential, naturally appear in this context. Although such functions are somewhat analytically exotic, the works of Ecalle, van der Hoeven and Schmeling show that some of them are amenable to geometric or formal descriptions. Hyperseries are an extension of transseries that can act as formal counterparts to more general germs including hyperexponentials. It turns out that hyperseries can be interpreted as Conway's surreal numbers. I will show that surreal numbers fill the gaps left in transseries, and I will explain how one can exploit this connection to endow the class of surreal numbers with a structure of field of hyperseries. |
