Appears in collection : 2018 - T1 - WS2 - Model theory and valued fields

A natural language to study valued fields is Ldiv := (+, −, ·, 0, 1, div) where div(x, y) is a binary predicate interpreted by v(x) ≤ v(y). An expansion (K,L) of (K,Ldiv) is C-minimal if for every elementary equivalent structure (K0,L), every L-definable subset of K0 is a Boolean combination of balls, in other words is quantifier free definable in the pure language Ldiv. A C-minimal valued field mustbe algebraically closed and conversely any pure algebraically closed non trivially valued field is C-minimal. We could hope to develop a theory of C-minimal valued fields on the model ofthat of o-minimal fields. In particular to develop a theory of C-minimal expansion ofthe valued field Cp on the model of o-minimal expansion of the real field. Analogies as well as serious obstructions appear. As an example any C-minimal expansion Cp of Cp is polynomially bounded, in contrast to the o-minimality of the real exponential field. On the other side, modulo a classical conjecture in o-minimality, any definable function in one variable definable in Cp is almost everywhere differentiable, as it happens in o-minimal fields. This is joint work with Pablo Cubides-Kovacsics.