In [1] it was shown that transition maps at non-resonant hyperbolic singularities of analytic vector fields in the plane are definable in an o-minimal structure. The proof uses the fact that the transition maps allow an asymptotic expansion and live in a quasianalytic class (see [2]). In the non-resonant case the asymptotic expansion is given by certain generalized power series. Based on this fact it was possible to define an appropriate generalization of this quasianalytic class to several variables and to develop a blow-up algorithm for resolution of singularities to obtain o-minimality in the spirit of previous work in o-minimal structures (see for example [3]). In the resonant case the asymptotic expansion is given by generalized power series where the monomials are additionally perturbed by logarithmic polynomials. The desired goal would be to develop a blow-up algorithm for resolution of singularities also in this case to obtain o-minimality of the transition maps in the resonant case and of related topics (see [4], [5]).But, the presence of logarithmic terms complicates the situation enormously. In this talk I discuss a possible approach towards a blow-up algorithm.

Literature:

[1]T.Kaiser, J.-P.Rolin and P.Speissegger: Transition maps at non-resonant hyperbolic singularities are o-minimal, Preprint, 2006,

arXiv:math/0612745, J. Reine Angew. Math. to appear

[2]Y. S. Ilyashenko, Finiteness theorems for limit cycles, Translations

of Mathematical Monographs 94 (1991)

[3]L.Van den Dries, and P. Speissegger: The real field with convergent

generalized power series, Trans. Amer. Math. Soc. 350 (1998), 4377--4421 [4]T. Kaiser: The Riemann mapping theorem for semianalytic domains and

o-minimality. Proc. Lond. Math. Soc. (3) 98, No. 2, 427-444 (2009)

[5]T.Kaiser: The Dirichlet problem in the plane with semianalytic raw

data, quasi analyticity, and o-minimal structure. Duke Math. J. 147, No.

2, (2009), 285-314.