Seminar

In joint work with Lou van den Dries and Joris van der Hoeven, we are investigating algebraic and model-theoretic properties of certain valued differential fields in which the valuation and derivation interact in a very strong way. Prime examples of interest are various fields of "transseries," first introduced independently by J. Écalle in his work on Hilbert’s 16th Problem and by the model theorists Dahn and Göring in their work around Tarski’s problem on real exponentiation, as well as Hardy fields (such as fields of germs of functions definable in an o-minimal expansion of the real field). One important question in this subject is what the right notion of "differentially henselian" should be. In this talk I will propose an answer to this question; I will introduce the concept of "newtonian" valued differential field and explain some of its fundamental properties. Lurking in the background is a version of the Newton polygon process for differential polynomials, which I also hope to describe.