Titre de la thèse: »Hyperséries et nombres surréels ».

Promoteurs de thèse: Madame Françoise Point, Monsieur Christian Michaux, Monsieur Joris Van Der Hoeven

Résumé: All regular growth rates can be represented as numbers, and vice versa. In this thesis, we propose to make this statement precise, and to give a proof.In the context of real geometry, regular functions appear in many different frameworks: continuous functions, analytic functions, quasi-analytic functions, germs in Hardy fields, definable maps in tame expansions of R. As we will soon explain, working with these notions poses difficulties when trying to account for all possible regular growth rates, because this entails the ability to show that certain operations, including derivation, integration, composition, summation, functional inversion, conjugation, etc., which naturally appear when studying these growth rates, preserve their defining features of continuity, quasi-analyticity.This is why it is sound to move away from the historically dependent notion of regular growth rates to consider instead a precise formal setting, in which objects act as functions with formal argument and values, and which is designed explicitly for its closure under many operations and equations. This is the realm of generalized power series. Defining derivations ∂: S àS and composition laws : S S^>R  S on an ordered field S of formal series containing the reals is a way to let those series act as regular, infinitely differentiable functions on S^>R. Transseries, as generalized power series based on operators exp, log and arithmetic operations, are a natural generalization of regular growth rate. As van der Hoeven’s PhD thesis illustrates, transseries are at the same time naturally closed under many operations and equations while being amenable to formal and algorithmic methods for solving equations or more general problems. Hyperseries, introduced in essence by Écalle and studied by van der Hoeven and Schmeling, are extensions of transseries with even stronger closure properties such as closure under conjugation. Van der Hoeven conjectured the existence of a large field Hy of hyperseries equipped with a derivation ∂ and a composition law such that any unary term t( ) in the first-order language over (Hy, +, , ∂, ) satisfies an following intermediate value property. We take such a field Hy as an ultimate field-with-no-escape“, taking all hyperseries in Hy to conveniently subsume all regular functions“.Now turning to the imprecise term number“, one may first be surprised by the idea that functions could be represented as as number. In fact this idea is already suggested in the work of du Bois-Reymond and Hardy where they manipulated growth rates as ordered quantities lying in a real-closed fields, implying that those quantities are amenable to the same arithmetic operations as real numbers are. Now the more surprising idea in this is the existence of a system of numbers that could account for all such quantities. Fortunately, candidate for this has been defined by Conway five decades ago, in the form of his class No of surreal numbers. As mathematicians took hold of the class No, it became a prominent candidate universal domain for several . Van der Hoeven conjectured that surreal numbers are canonically isomorphic to Hy, the isomorphism being an evaluation map sending each hyperseries ƒ ∈ Hy to its value at ω“ ƒ (ω) ∈ Hy(ω) = No.The main goal of this thesis is to prove this last conjecture, taken as a precise formulation of the introductory analogy. We do this by representing surreal numbers as hyperseries, i.e. as formal expressions involving a definite set of operations, of which we will provide a solid understanding. On the class of surreal numbers, we will see how to define (besides usual arithmetic operations) an infinite arity summation operator , exponentials exp and logarithms log and so-called transfinite iterators exp_α and log_α thereof, where α ranges in the class On of ordinals. We will prove that such operators allow for a full representation of numbers as hyperseries. In doing so, we construe a field Hy(ω) of hyperseries in ω, thus answering van der Hoeven’s conjecture in the positive.