Principal Kinematic Formulas for Germs of Closed Definable Sets

Some results of geometry over Henselian fields with analytic structure

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…

Local metric properties and regular stratifications of p-adic definable sets

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our field and show, as it is the case in the real setting, that, up to some multiplicities, the local density may be computed on this distinguished tangent cone.We also prove that these distinguished tangent…

二维中的多重$\mathcal{K}$-Lipschitz等价性

Multi-$\mathcal{K}$-Lipschitz equivalence in dimension two

本文通过引入Multipizza划分将平面多重芽转化为非阿基米德空间，证明了其在多重$\mathcal{K}$-Lipschitz等价下的存在性、唯一性与完全不变性。

Stable domination and independence in algebraically closed valued fields

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stabl…

Equi-singularity of real families and Lipschitz Killing curvature densities at infinity

Fix an o-minimal structure expanding the ordered field of real numbers. Let $(W_y)_{y\in\mathbb{R}^s}$ be a definable family of closed subsets of $\mathbb{R}^n$ whose total space $W = \cup_y W_y\times y$ is a closed connected $C^2$ definable sub-manifold of $\mathbb{R}^n\times\mathbb{R}^s$. Let $φ:W \to\mathbb{R}^s$ be the restriction of the projection to the second factor. After defining $K(φ)$, the set of generalized critical values of $φ$, showing that they are closed and definable of positive codimension in $\mathbb{R}^s$, contain the bifurcation values of $φ$ and are stable under generic …

Oscillating integrals and Newton polyhedra

We establish a principal value integral formula, for the residue of the largest non-trivial candidate pole of the real or complex local zeta function associated to an analytic germ f, which is non-degenerate with respect to its Newton polyhedron. In particular, up to an easy non-zero factor, this residue only depends on the (tau_0)-principal part of f, where tau_0 is the smallest face of the Newton polyhedron intersecting the diagonal. This formula allows us to prove some vanishing results for the residue. More precisely, we prove that the residue vanishes when tau_0 is unstable, and we give a…