H-minimality (with R. Cluckers, I. Halupczok)

The development and numerous applications of strong minimality and later o-minimality has given serious credit to the general model theoretic idea that imposing strong restrictions on the complexity of arity one sets in a structure can lead to a rich tame geometry in all dimensions. O-minimality (in an ordered field), for example, requires that subsets of the affine are finite unions of points and intervals.

In this talk, I will present a new minimality notion (h-minimality), geared towards henselian valued fields of characteristic zero, generalising previously considered notions of minimality for valued fields (C,V,P …) that does not, contrary to previously defined notions, restrict the possible residue fields and value groups. By analogy with o-minimality, this notion requires that definable sets of of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading us to a whole family of notions of h-minimality. I will then describe consequences of h-minimality, among which the jacobian property that plays a central role in the development of motivic integration, but also various higher degree and arity analogs.