Submissions (2)
Accepted (2):
Composition of transseries, monotonicity, and analyticity — Vincenzo Mantova <v.l.mantova@leeds.ac.uk>
Transseries generalise power series by including exponential and logarithmic terms, if not more, and can be interpreted as germs of a non-standard Hardy field by composition (for instance, on surreal numbers). I'll discuss a few results that must 'obviously' be true, yet their proofs are not obvious: that composition is monotonic in both arguments, only proved by Edgar for LE-series, that it satisfies a suitable Taylor theorem and that in fact composition is 'analytic with large radius of convergence' (joint with V. Bagayoko), something which appeared before in various special forms, but not in full generality. I'll discuss briefly what I cannot prove yet (convexity!). I'll show how monotonicity and Taylor can be used to prove some fairly general normalisation results for hyperbolic transseries (joint with D. Peran, J.-P. Rolin, T. Servi).
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Multisummability relative to certain quasianalytic classes realted to Dulac's Problem — Patrick Speissegger <speisse@mcmaster.ca>
Using Tougeron’s characterization of multisummable series (in the positive real direction), 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 series (again in the positive real direction). This class is shown to generate an o-minimal expansion $\mathbb{R}_{\mathcal{G}^*}$, whose expansion by the exponential function then defines the restrictions to some unbounded interval of both the Gamma and zeta functions. More recently, with Ilgwon Seo, we have 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.
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