Some results of geometry over Henselian fields with analytic structure

TL;DR

This paper extends geometric results over Henselian valued fields with analytic structures, establishing properties like piecewise continuity, uschiewicz inequalities, and curve selection, using advanced model-theoretic tools and resolution techniques.

Contribution

It generalizes previous algebraic results to broader analytic settings over Henselian fields, including local rings and definable functions, via new geometric and model-theoretic methods.

Findings

Piecewise continuity of definable functions

Versions of uschiewicz inequality for definable functions

Curve selection and Hf6lder continuity results

Abstract

The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is a complete rank one valued fields with the Tate algebra of strictly convergent power series. The algebraic case was treated in our previous papers. Here we are going to carry over the research to the general analytic settings. Also considered are the local rings of analytic function germs induced by a given separated Weierstrass system, which turn out to be excellent and regular. Several results are established as, for instance, piecewise continuity of definable functions, several versions of the \L{}ojasiewicz inequality, H\"{o}lder continuity of definable functions continuous on closed bounded subsets of the affine space and curve selection for definable sets. Likewise as before, at the center of our approach is the closedness theorem to the effect that every projection with closed bounded fiber is a definably closed map. It enables application of resolution of singularities and of transformation to a normal crossing by blowing up (here applied to the local rings of analytic function germs) in much the same way as over locally compact ground fields. We rely on elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers--Lipshitz--Robinson, as well as on relative quantifier elimination for ordered abelian groups, due to Cluckers--Halupczok. Besides, other two ingredients of the proof of the closedness theorem are existence of the limit (after finite partitioning of the domain) for a definable function of one variable and fiber shrinking, being a relaxed version of curve selection.