Nous associons à tout polydisque compact B [appartenant à] Rn une algèbre CB de fonctions réelles de classe C∞ définies au voisinage de B. La collection des algèbres CB est supposée stable par certaines opérations, dont la composition et la dérivation partielle. Nous supposons de plus que, lorsque B est centrée à l’origine, l’algèbre des germes à l’origine des éléments de CB est quasianalytique (c’est à dire qu’elle ne contient pas de germe plat). A l’aide de ces fonctions, nous définissons des ensembles C-semi- analytiques et C-sous-analytiques comme on le fait traditionnellement en géométrie analytique réelle. Notre résultat principal est un théorème du type Tarski-Seidenberg pour ces ensembles. Son énoncé dit essentiellement que les ensembles sous-C-analytiques peuvent être définis par des égalités et des inégalités satisfaites par des termes obtenus en composant des fonctionsdes algèbres C_B , les fonctions x → x1/n , et la fonction x → 1/x. Sa preuve se fait en exprimant les solutions de sytèmes d’équations quasianalytiques au moyen d’un théorème de préparation issu de la théorie des modèles / We associate to every compact polydisk B [belonging to ] Rn an algebra CB of real functions defined in a neighborhood of B. The collection of these algebras is supposed to be closed under several operations, such as composition and partial derivatives. Moreover, if the center of B is the origin, we assume that the algebra of germs at the origin of elements of CB is quasianalytic (it does not contain any flat germ). We define with these functions the collection of C-semianalytic and C-subanalytic sets according to the classical process in real analytic geometry. Our main result is an analogue of Tarski-Seidenberg's usual result for these sets. It says that the sub-C-subanalytic sets may be described by means of equalities and inequalities by terms obtained by composition of elements of the algebras CB, the functions x->^{1/n} and the function x->1/x. It is proved via a model theoretic preparation theorem

<p>Studying properties of random polynomials has marked a shift in algebraic geometry. Instead of worst-case analysis, which often leads to overly pessimistic perspectives, randomness helps perform average-case analysis, and thus obtain a more realistic view. Also, via Erdos' astonishing 'probabilistic method', one can potentially obtain deterministic results by introducing randomness into a question that apriori had nothing to do with randomness. </p> <p><br></p> <p>In this thesis, we study topological questions in real algebraic geometry, o-minimal geometry and random algebraic geometry, with motivation from incidence combinatorics. Specifically, we prove results along two different threads:</p> <p><br></p> <p>1. Topology of semi-algebraic and definable (over any o-minimal structure over R) sets, in both deterministic and random settings.</p><p>2. Topology of random hypersurface arrangements. In this case, we also prove a result that could be of independent interest in random graph theory.</p> <p><br></p> <p>Towards the first thread, motivated by applications in o-minimal incidence combinatorics, we prove bounds (both deterministic and random) on the topological complexity (as measured by the Betti numbers) of general definable hypersurfaces restricted to algebraic sets. Given any sequence of hypersurfaces, we show that there exists a definable hypersurface G, and a sequence of polynomials, such that each manifold in the sequence of hypersurfaces appears as a component of G restricted to the zero set of some polynomial in the sequence of polynomials. This shows that the topology of the intersection of a definable hypersurface and an algebraic set can be made <i>arbitrarily pathological</i>. On the other hand, we show that for random polynomials, the Betti numbers of the restriction of the zero set of a random polynomial to any definable set deviates from a Bezout-type bound with <i>bounded probability</i>.</p> <p><br></p> <p>Progress in o-minimal incidence combinatorics has lagged behind the developments in incidence combinatorics in the algebraic case due to the absence of an o-minimal version of the Guth-Katz <i>polynomial partitioning</i> theorem, and the first part of our work explains why this is so difficult. However, our average result shows that if we can prove that the measure of the set of polynomials which satisfy a certain property necessary for polynomial partitioning is suitably bounded from below, by the <i>probabilistic method</i>, we get an o-minimal polynomial partitioning theorem. This would be a tremendous breakthrough and would enable progress on multiple fronts in model theoretic combinatorics. </p> <p><br></p> <p>Along the second thread, we have studied the average Betti numbers of <i>random hypersurface arrangements</i>. Specifically, we study how the average Betti numbers of a finite arrangement of random hypersurfaces grows in terms of the degrees of the polynomials in the arrangement, as well as the number of polynomials. This is proved using a random Mayer-Vietoris spectral sequence argument. We supplement this result with a better bound on the average Betti numbers when one considers an <i>arrangement of quadrics</i>. This question turns out to be equivalent to studying the expected number of connected components of a certain <i>random graph model</i>, which has not been studied before, and thus could be of independent interest. While our motivation once again was incidence combinatorics, we obtained the first bounds on the topology of arrangements of random hypersurfaces, with an unexpected bonus of a result in random graphs.</p>

<p>The main subject of investigation is the so-called "immediate expansion''<br />phenomenon in various first-order valued-field structures over the<br />corresponding underlying field structures. In particular, certain "valued<br />o-minimal fields'', certain Henselian valued fields with non-divisible valued<br />groups, and certain separably closed valued fields of finite imperfection degree, are<br />shown to have this property.</p> / Doctor of Philosophy (PhD)

L'objectif de la thèse est l'étude des propriétés géométriques des ensembles définissables dans les structures o-minimales et de ses applications. Il existe trois principaux résultats présentés dans cette thèse. Le premier est une preuve géométrique de l'existence de stratifications vérifiant les conditions (a) et (b) de Whitney d'ensembles définissables. Ce résultat fut d'abord prouvé par T. L. Loi en 1994 par une autre méthode. Le second est une preuve de l'existence de stratifications de Lipschitz (dans le sens de Mostowski) pour les ensembles définissables dans une structure o-minimale polynomialement bornée. Ceci est une généralisation de résultats de Parusin'ski en 1994 pour les ensembles sous-analytiques. Le troisième résultat est au sujet de la continuité des variations de géométrie intégrale appelées courbures de Lipschitz Killing locales, qui ont été introduites par A. Bernig et L. Broker en 2002. Nous prouvons que les courbures de Lipschitz Killing locales sont continues le long de strates de stratifications de Whitney d'ensembles définissable dans une structure o-minimale polynomialement bornée, et si les stratifications sont (w) régulières alors les courbures de Lipschitz Killing locales sont localement lipschitziennes le long des strates. / The thesis focus on study geometric properties of definable sets in o-minimal structures and its applications. There are three main results presented in this thesis. The first is a geometric proof of the existence of Whitney (a) and (b)-regular stratifications of definable sets. The result was initially proved by T. L. Loi in 1994 by using another method. The second is a proof of existence of Lipschitz stratifications (in the sense of Mostowski) of definable sets in a polynomially bounded o-minimal structure. This is a generalization of Parusinski's 1994 result for subanalytic sets. The third result is about the continuity of of variations of integral geometry called local Lipschitz Killing curvatures which were introduced by A. Bernig and L. Broker in 2002. We prove that Lipschitz Killing curvatures are continuous along strata of Whiney stratifications of definable sets in a polynomially bounded o-minimal structure. Moreover, if the stratifications are (w)-regular the Lipspchitz Killing curvatures are locally Lipschitz.