Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

     

Flattening and Subanalytic Sets in Rigid Analytic Geometry

Let K be an algebraically closed field endowed with a completenon-archimedean norm with valuation ring R. Let f: Y X be amap of K-affinoid varieties. In this paper we study the analyticstructure of the image f(Y) X; such an image is a typical exampleof a subanalytic set. We show that the subanalytic sets areprecisely the D-semianalytic sets, where D is the truncateddivision function first introduced by Denef and van den Dries.This result is most conveniently stated as a Quantifier Eliminationresult for the valuation ring R in an analytic expansion ofthe language of valued rings. To prove this we establish a Flattening Theorem for affinoidvarieties in the style of Hironaka, which allows a reductionto the study of subanalytic sets arising from flat maps, thatis, we show that a map of affinoid varieties can be renderedflat by using only finitely many local blowing ups. The caseof a flat map is then dealt with by a small extension of a resultof Raynaud and Gruson showing that the image of a flat map ofaffinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in thezero characteristic case, a Uniformization Theorem for subanalyticsets: a subanalytic set can be rendered semianalytic using onlyfinitely many local blowing ups with smooth centres. As a corollarywe obtain the fact that any subanalytic set in the plane R2is semianalytic. 2000 Mathematical Subject Classification: 32P05,32B20, 13C11, 12J25, 03C10.     

Flattening and analytic continuation of affinoid morphisms: remarks on a paper of Gardener and Schoutens

We give an example of an affinoid curve without analytic continuation.We use this to produce an example of an affinoid morphism thatcannot be flattened by a finite sequence of local blow-ups.Thus the global rigid analogue of Hironaka's complex analyticflattening theorem given by T. Gardener and H. Schoutens, inTheorem 2.3 of ‘Flattening and subanalytic sets in rigidanalytic geometry’, Proc. London Math. Soc (3) 83 (2001)681–707, is not true. Since this is a key step in theproof of the affinoid elimination theorem (loc. cit. Theorem3.12), that proof contains a serious gap. We also give an exampleof an affinoid subset of the plane that is not the image undera proper rigid analytic map of a set that is globally semianalyticin the domain of that map. This clarifies the relationship amongseveral natural categories of rigid subanalytic sets.2000 MathematicsSubject Classification 32P05 (primary), 03C10, 32B20 (secondary).     

Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .      

Quantifier elimination for the theory of algebraically closed valued fields with analytic structure

The theory of algebraically closed non‐Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz, and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper. This theorem also has other consequences in the geometry of definable sets. The method of proving quantifier elimination in this paper for an analytic language does not require the algebraic quantifier elimination theorem of Weispfenning, unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mycielski ideal and the perfect set theorem

We make several observations on the Mycielski ideal and prove a version of the perfect set theorem concerning this ideal for analytic sets: If is an analytic set all projections of which are uncountable, then there is a perfect set a projection of which is the whole space. We also prove that (a modification of) an infinite game of Mycielski is determined for analytic sets.

     

On Local Convergence of the Method of Alternating Projections

The method of alternating projections is a classical tool to solve feasibility problems. Here we prove local convergence of alternating projections between subanalytic sets \(A,B\) under a mild regularity hypothesis on one of the sets. We show that the speed of convergence is \({\mathcal {O}}(k^{-\rho })\) for some \(\rho \in (0,\infty )\).     

Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu’s result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.     

The structure and maximum number of maximum independent sets in trees

A subset of vertices is a maximum independent set if no two of the vertices are joined by an edge and the subset has maximum cardinality. in this paper we answer a question posed by Herb Wilf. We show that the greatest number of maximum independent sets for a tree of n vertices is We give the families of trees on which these maxima are achieved. Proving which trees are extremal depends upon the structure of maximum independent sets in trees. This structure is described in terms of adjacency rules between three types of vertices, those which are in all, no, or some maximum independent sets. We show that vertices that are in some but not all maximum independent sets of the tree are joined in pairs by the α-critical edges (edges whose removal increases the size of a maximum independent set). The number of maximum independent sets is shown to depend on the structure within the tree of the α-critical edges.     

Smooth points ofp-adic subanalytic sets

Ap-adic subanalytic set shares with a real subanalytic set the fundamental property that its singular locus is itself subanalytic. Furthermore, given ap-adic subanalytic function ? with domain contained in ? _p ^m , there is an integerL such that for any pointx ₀ ∈ ? _p ^m in a neighborhood of whichf is defined,f has a Taylor approximation up to orderL atx ₀ if, and only if, ? is analytic aroundx ₀. These results extend to thep-adic fields real variables theorems by M. Tamm [21].     

Control of radii of convergence and extension of subanalytic functions

Let : denote a real analytic function on an open subset of , and let denote the points where does not admit a local analytic extension. We show that if is semialgebraic (respectively, globally subanalytic), then is semialgebraic (respectively, subanalytic) and extends to a semialgebraic (respectively, subanalytic) neighbourhood of . (In the general subanalytic case, is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series centred at points in the image of an analytic mapping , in terms of the radii of convergence of at points , where denotes the Taylor expansion of at .

     

Graph Products,Fourier Analysis and Spectral Techniques

We consider powers of regular graphs defined by the weak graph product and give a characterization of maximum-size independent sets for a wide family of base graphs which includes, among others, complete graphs, line graphs of regular graphs which contain a perfect matching and Kneser graphs. In many cases this also characterizes the optimal colorings of these products.We show that the independent sets induced by the base graph are the only maximum-size independent sets. Furthermore we give a qualitative stability statement: any independent set of size close to the maximum is close to some independent set of maximum size.Our approach is based on Fourier analysis on Abelian groups and on Spectral Techniques. To this end we develop some basic lemmas regarding the Fourier transform of functions on generalizing some useful results from the case.     

Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique

Summary We study the structure of the one dimensional analytic quasi-compact spaces over a complete non archimedean valued field. An affinoid open subset U of a one dimensional analytic quasi-compact space X is defined by a meromorphic function f on X;i.e. U is the set of all x in X such that f is holomorphic at x and ¦f(x)¦1.The set of the meromorphic functions on X which are holomorphic on U is dense in the ring of all holomorphic functions on U. An irreducible, one dimensional quasi-compact space is either affinoid, or projective. An analytic reduction of X is defined by a meromorphic invertible function f on X;i.e. the reduction is isomorphic to the reduction associated to the covering ¦f(x)¦1and ¦f(x)¦1.     

Rigid cantor sets in with simply connected complement

We prove that there exist uncountably many inequivalent rigid wild Cantor sets in with simply connected complement. Previous constructions of wild Cantor sets in with simply connected complement, in particular the Bing- Whitehead Cantor sets, had strong homogeneity properties. This suggested it might not be possible to construct such sets that were rigid. The examples in this paper are constructed using a generalization of a construction of Skora together with a careful analysis of the local genus of points in the Cantor sets.

     

The planar Cantor sets of zero analytic capacity and the local -Theorem

In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the -Theorem.

     

Orderings and maximal ideals of rings of analytic functions

We prove that there is a natural injective correspondence between the maximal ideals of the ring of analytic functions on a real analytic set and those of its subring of bounded analytic functions. By describing the maximal ideals in terms of ultrafilters we see that this correspondence is surjective if and only if is compact. This approach is also useful for studying the orderings of the field of meromorphic functions on .

     

Rigid pentagons in hypercubes

Two sets of vertices of a hypercubes in ⁿ and ^m are said to be equivalent if there exists a distance preserving linear transformation of one hypercube into the other taking one set to the other. A set of vertices of a hypercube is said to be weakly rigid if up to equivalence it is a unique realization of its distance pattern and it is called rigid if the same holds for any multiple of its distance pattern. A method of describing all rigid and weakly rigid sets of vertices of hypercube of a given size is developed. It is also shown that distance pattern of any rigid set is on the face of convex cone of all distance patterns of sets of vertices in hypercubes.Rigid pentagons (i.e. rigid sets of size 5 in hypercubes) are described. It is shown that there are exactly seven distinct types of rigid pentagons and one type of rigid quadrangle. It is also shown that there is a unique weakly rigid pentagon which is not rigid. An application to the study of all rigid pentagons and quadrangles inL ¹ having integral distance pattern is also given.This work was done during a visit of both the authors to Mehta Research Institute, Allahabad, India.     

Formule de Cauchy-Crofton pour la densité des ensembles sous-analytiques

For the density of subanalytic sets, we give a formula analogous to the classical Cauchy-Crofton formula for the volume. To do this we define the local polar profiles of a subanalytic set and the corresponding finite sequence of local multiplicities which is the real counterpart of the multiplicity of complex analytic sets.     

A trace formula for rigid varieties,and motivic Weil generating series for formal schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of f at x. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When is the formal completion of f at a closed point x of the special fiber , we obtain the local motivic zeta function of f at x. The research for this article was partially supported by ANR-06-BLAN-0183.