Person: Speissegger, Patrick
Analytic continuations of log-exp-analytic germs
2019, Kaiser, Tobias, Speissegger, Patrick
We describe maximal, in a sense made precise, L-analytic continuations of germs at +∞ of unary functions definable in the o-minimal structure Ran,exp on the Riemann surface L of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie’s theorem on definable complex analytic continuations of germs belonging to the residue field Rpoly of the valuation ring of all polynomially bounded definable germs.
Quasianalytic Ilyashenko algebras
2018-02-01, Speissegger, Patrick
I construct a quasianalytic field F of germs at +∞ of real functions with logarithmic generalized power series as asymptotic expansions, such that F is closed under differentiation and log-composition; in particular, F is a Hardy field. Moreover, the field F o (−log) of germs at 0+ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.