Special Year Research Seminar

Using Tougeron’s characterization of multisummable (in the positive real directions) series, the latter can be viewed as infinite series of convergent power series with radii of convergence shrinking to 0. In joint work with Jean-Philippe Rolin and Tamara Servi we showed that, if we replace “convergent power series” with “convergent generalized power series”, we obtain a larger class of multisummable (in the positive real directions) series. This class is shown to generate an o-minimal expansion $R_G*$, whose expansion by the exponential function then defines the restrictions to some unbounded interval of both the Gamma and zeta functions. More recently, Ilgwon Seo has been further generalizing this construction by replacing “convergent power series” with “almost regular generalized power series”. The resulting Hardy field is a first step towards filling the remaining gap in Ilyashenko’s proof of Dulac’s problem.