Considerations on the asymptotic behavior and the growth at infinity of functions are often used to solve ODEs. The field of transseries was first defined by Dahn and Göring and by Ecalle and Il'yashenko independently in the 80s and 90s. It is a generalization of the germs at infinity of exp-log functions. Transseries formalize the so called exp-log growth class in a setting with more algebraic operations than germs enabling more computational power. Work from J van der Hoeven (1997) has shown that one can solve polynomial differential equations in the field of grid based transseries using setting-specific techniques. In 2008, M. Aschenbrenner, L. van der Dries and J. van der Hoeven have proved that the theory of transseries is recursively axiomatisable and admits QE in a finite language containing the field operations and the derivation. Therefore since differential polynomials over the transseries are definable in that langage there exists an algorithm solving polynomial ODE over any model of the theory. Our aim is to present a Preparation theorem in the style of Weierstrass' such that given any differential polynomial P we construct a cell decomposition of the space such that P rewrites on each cell as a simple product of factors from which the roots of P are easily readable : this is equivalent to eliminating quantifiers from a certain type of formula.
In this talk I will describe and illustrate the field of transseries without going into the details of its technical construction, before presenting a preparation theorem for polynomials, our conjecture and various examples from our work. |