Speissegger, Patrick

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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.

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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.