Tamara Servi (IMJ-PRG & Fields Institute)

Interdefinability and compatibility in certain o-minimal expansions of the real field

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The sets definable in an o-minimal expansion of the real field have a tame topological behaviour (uniform finiteness, good dimension theory, no pathological phenomena). Being able to tell if a certain real set or function is definable in a given o-minimal structure gives us information on how tame the geometry of that object is.

Let us say that a real function $f$ is o-minimal if the expansion $(R,f)$ of the real field $R$ by $f$ is o-minimal. A function $g$ is definable from $f$ if $g$ is definable in $(R,f)$. Two o-minimal functions $f$ and $g$ are compatible if $(R,f,g)$ is o-minimal. I will discuss the o-minimality, the interdefinability and the compatibility of two special functions, Euler’s Gamma and Riemann’s Zeta, restricted to the reals. Joint work with J.-P. Rolin and P. Speissegger.