Mild Parametrization in O-Minimal Structures

Derya Çiray (University of Konstanz)

Frank Adams 1,

Semi-algebraic sets are subsets of \(\mathbb{R}^n\) described by finitely many polynomial equalities and inequalities. They have nice geometrical properties and are stable under finite unions, products, taking complements and projections (Tarski-Seidenberg Theorem). An o-minimal structure is an axiomatic framework that allows to extend some properties of semialgebraic geometry to a non-algebraic setting.
The application of mild parametrization, which is a smooth parametrization with some control on the derivatives, was first introduced in the realm of diophantine geometry by J. Pila, to obtain results about the density of rational points on the graphs of non-algebraic pfaffian functions. Furthermore he has shown that mild parametrization with sufficient uniformity in parameters would be sufficient to establish Wilkie's conjecture which is about density of rational points of the o-minimal structure expanded by the exponential function.
In this talk I will talk about interactions between o-minimal structures and mild parametrization and discuss whether certain o-minimal expansions of the reals admit mild parametrization
or not.

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