Some complex analysis for polynomially bounded o-minimal structures:

Abstract: We consider complex analytic functions on the unit disk which
are definable (with parameters) in an elementary extension of a
polynomially bounded, o-minimal expansion of the real field. We show that
such functions cannot distort shape by very much (a fact that is not true
without the polynomially boundedness assumption), and use this to define a
pregeometry capturing the notion of "definably complex analytic over". If
time permits I will go on to discuss a complex version of the valuation
inequality.