Georges Comte

Tame geometry with application in smooth analysis

Preface.- Introduction and Content.- Entropy.- Multidimensional Variations.- Semialgebraic and Tame Sets.- Some Exterior Algebra.- Behavior of Variations under Polynomial Mappings.- Quantitative Transversality and Cuspidal Values for Polynomial Mappings.- Mappings of Finite Smoothness.- Some Applications and Related Topics.- Glossary.- References.

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 cones stabilize for small enough subgroups. We finally obtain the p-adic counterpart of the Cauchy-Crofton formula for the density. To prove these results we use the Lipschitz decomposition of definable p-adic sets of (5) and prove here the genericity of the regularity conditions for stratification such as .wf/, .w/, .af/, .b/ and .a/ conditions. Mathematics Subject Classification (2010). Primary 03C10, 03C98, 12J10, 14B05, 32Sxx; Secondary 03C68, 11S80 14J17.

Grothendieck ring of semialgebraic formulas and motivic real Milnor fibers

— We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows to express a class as a Z[ 1 2 ]linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincaré polynomial for basic semialgebraic formulas. We then define zeta functions with coefficients in our ring, built on semialgebraic formulas in arc spaces. We show that they are rational and relate them to the topology of real Milnor fibres.

On Zariski's multiplicity problem

We show that to answer affirmatively Zariskis question concerning the topological invariance of the multiplicity of complex analytic hypersurfaces at isolated singular points, it suffices to prove two combined statements, each of which may be obtained separately.

7. Behaviour of Variations under Polynomial Mappings

We study here the multidimensional variations of the image under a polynomial mapping of a semialgebraic set. We bound from above the i-th variation of the image by the i-th variation of the set and by the i-th Jacobian. This allows us to prove the quantitative Sard theorem for polynomial functions. We also define and study the “variations” of a polynomial mapping, and we finally bound from below the variation of the image.

5. Variations of Semialgebraic and Tame Sets

We study here multidimensional variations of semialgebraic and tame sets, partly following [Vit 1], [Iva 1]. The stress is lain on properties which distinguish tame sets, in particular, correlations between variations of \(\epsilon\)-neighborhood, comparison of variations of two sets with a small Hausdorff distance etc...

9. Mappings of Finite Smoothness

We prove the quantitative Morse-Sard theorem for \(\mathcal{C}^k\) mappings with n variables, i.e. we bound the \(\epsilon\)-entropy of near-critical values. In particular, we give, for the entropy dimension of the rank-\(\nu\) set of critical values, a bound depending only on n, \(\nu\) and k. We then give examples showing that our statement is the best possible. We also give the \(\mathcal{C}^k\) version of the polynomial quantitative transversality of Chapter 8.

4. Semialgebraic and Tame Sets

We prove in this chapter a classical result: the number of connected components of a plane section \(\mathrm{P}\cap \mathrm{A}\) of a semialgebraic set \(\mathrm{A}\) is uniformly bounded with respect to \(\mathrm{P}\). An explicit bound is given in terms of the diagram of \(\mathrm{A}\) and the dimension of \(\mathrm{P}\). We give a construction which provides a semialgebraic section of bounded complexity for any polynomial mapping of semialgebraic sets. In particular, any two points in a connected semialgebraic set can be joined by a semialgebraic curve of bounded complexity. We also give the definition of an o-minimal structure on the real field and show that in such a category the uniform bound for the number of connected components of plane sections holds.

6. Some Exterior Algebra

We give in this chapter some basic definitions and well-known results in exterior algebra, in order to get a convenient definition of a size for differentials of mappings. The behaviour of this size under projections and restrictions to subspaces (as required by the variations approach) is studied.

3. Multidimensional Variations

We define in this chapter the multidimensional variations, study their properties and show how the \(\epsilon\)-entropy of a subset \(A\) of \(\mathbb{R}^n\) can be bounded in terms of variations of \(A\). This form one of the main technical tools used in this book.