5 may 2015 -- 16:00
Aula Riunioni, DM, Pisa
Seminari di o-minimalità, Pisa
In order to understand the local behaviour of the maps definable in expansions of the complex field C by holomorphic functions, and since, identifying C with R2, these functions are locally definable in reducts of the structure Ran generated by all subanalytic sets, Wilkie started a systematic study of the reducts of Ran generated by a collection of restricted holomorphic maps.
He formulated the following conjecture. Let A be a familly of holomorphic maps, and denote by RA
the (o-minimal) expansion of the real field R by the real and imaginary parts of the functions in A, restricted to the relatively compact boxes of their domain. Conjecture (Wilkie 08) : the holomorphic maps which are locally definable in RA
are all obtained from the collection of all functions in A and all polynomials by closing such a collection under composition, Schwartz reflection, partial derivation and taking implicit functions. We give three counterexample to this conjecture, each of which shows that another operation, issued from the process of resolution of singularity, is needed to get all locally definable functions. Namely, adding monomial division, composition with nth-roots, and blow-downs is needed to describe all locally definable holomorphic maps. This is a joint work with G. Jones, J. Kirby and T. Servi.