This paper examines the theory of an adelization of Shintani's zeta function, especially as it relates to density theorems for discriminants of cubic extensions of number fields. / Mathematics

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The motivating question behind this body of research is Smale’s 17th problem: Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? While certain aspects and viewpoints of this problem have been solved, the analog over the real numbers largely remains open. This is an important question with applications in celestial mechanics, kinematics, polynomial optimization, and many others. Let A = {α_1, . . . , α_n+k } ⊂ Zn. The A-discriminant variety is, among other things, a tool that can be used to categorize polynomials based on the topology of their real solution set. This fact has made it useful in solving aspects and special cases of Smale’s 17th problem. In this thesis, we take a closer look at the structure of the A-discriminant with an eye toward furthering progress on analogs of Smale’s 17th problem. We examine a mostly ignored form called the Horn uniformization. This represents the discriminant in a compact form. We study properties of the Horn uniformization to find structural properties that can be used to better understand the A-discriminant variety. In particular, we use a little known theorem of Kapranov limiting the normals of the A-discriminant amoeba. We give new O(n^2) bounds on the number of components in the complement of the real A-discriminant when k = 3, where previous bounds had been O(n^6) or even exponential before that. We introduce new tools that can be used in discovering various types of extremal A-discriminants as well as examples found with these tools: a family of A-discriminant varieties with the maximal number of cusps and a family that appears to asymptotically admit the maximal number of chambers. Finally we give sage code that efficiently plots the A-discriminant amoeba for k = 3. Then we switch to a non-Archimedean point of view. Here we also give O(n^2) bounds for the number of connected components in the complement of the non- Archimedean A-discriminant amoeba when k = 3, but we also get a bound of O(n^(2(k−1)(k−2)) )when k > 3. As in the real case, we also give a family exhibiting O(n^2) connected components asymptotically. Finally we give code that efficiently plots the p-adic A-discriminant amoeba for all k ≥ 3. These results help us understand the structure of the A-discriminant to a degree, as yet, unknown. This can ultimately help in solving Smale’s 17th problem as it gives a better understanding of how complicated the solution set can be.

Il existe de nombreuses situations où des objets géométriques ou topologiques (comme les configurations de points du plan, les applications lisses entre variétés, les hypersurfaces projectives complexes) sont paramétrés par des éléments d'un espace vectoriel. Un discriminant (généralisé) est un sous-ensemble formé des éléments singuliers (dans un sens à préciser) d'un tel espace vectoriel. Par la dualité d'Alexander, les groupes de cohomologie du complémentaire d'un discriminant sont isomorphes aux groupes d'homologie de Borel-Moore du discriminant même. Souvent, ces derniers groupes peuvent être calculés en utilisant une certaine résolution naturelle du faisceau constant sur le discriminant ; par référence à leur construction, ces <br />résolutions sont parfois appelées coniques.<br /><br />Dans cette thèse, nous généralisons la méthode des résolutions coniques qui a été proposée par V. A. Vassiliev afin d'étudier la cohomologie des espaces des hypersurfaces projectives lisses complexes. Notre construction se base sur les relations d'inclusion entre les lieux singuliers plutôt qu'entre les systèmes linéaires correspondants. Cela nous permet d'effectuer certains calculs qui semblent être hors de portée de l'approche originelle. Pour illustrer notre méthode, nous calculons la cohomologie rationnelle de l'espace des courbes lisses complexes planes de degré 5, de l'espace des courbes bielliptiques lisses sur une quadrique non dégénérée dans l'espace projectif complexe de dimension 3, ainsi que de l'espace des courbes cubiques réelles lisses planes.<br /><br />La thèse contient un appendice où l'on démontre le résultat suivant. Supposons que le cercle est muni d'un atlas où tous les changements de cartes sont des homographies ; alors ce cercle borde une surface orientable munie d'un atlas où tous les changements de cartes sont aussi des homographies (à coefficients<br />complexes cette fois-ci) et sont compatibles dans le sens évident avec les applications de changement de cartes sur le bord. Dans l'appendice, nous montrons également que la classification des structures projectives sur le cercle donnée il y a longtemps par N. Kuiper n'est pas tout à fait correcte, et nous complétons cette classification.

Cette thèse traite du comptage d'extensions cubiques relatives. Dans le premier chapitre on traite un travail commun avec Henri Cohen. Soit k un corps de nombres. On donne une formule asymptotique pour le nombre de classes d'isomorphisme d'extensions cubiques L/k telles que la clôture galoisienne de L/k contienne une extension quadratique fixée K_2/k. L'outil principal est la théorie de Kummer. Dans le second chapitre, on suppose k un corps quadratique imaginaire (avec nombre de classes 1) et on décrit un algorithme pour énumerer toutes les classes d'isomorphisme d'extensions cubiques L/k jusqu'à une certaine borne X sur la norme du discriminant relatif.

The classical discriminant D_n of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for D_n. In particular, all of the formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Second, for the discriminant of the polynomial x^4+a_2x^2+a_3x+a_4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg a_i=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E_6 singularity {x^4-y^3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

The classical discriminant D_n of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for D_n. In particular, all of the formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Second, for the discriminant of the polynomial x^4+a_2x^2+a_3x+a_4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg a_i=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E_6 singularity {x^4-y^3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

One of the central themes of modern Number Theory is to study properties of Galois and automorphic representations and connections between them. In our dissertation, we describe two different projects that study properties of these objects. In our first project, which is analytic in nature, we consider Artin representations of Q of dimension 3 that are self-dual. We show that these occur with density 0 when counted using the conductor. This provides evidence that self-dual representations should be rare in all dimensions. Our second project, which is more algebraic in nature, is related to automorphic representations. We show the existence of canonical models for certain unitary Shimura varieties. This should help us in computing certain cohomology groups of these varieties, in which regular algebraic automorphic representations having useful properties should be found.

Submitted by (lucia.rodrigues@ufrpe.br) on 2017-03-29T13:33:25Z No. of bitstreams: 1 Fabiano Neves Nader.pdf: 355299 bytes, checksum: 1b2a2621a49984f5afa2c5660f9b322a (MD5) / Made available in DSpace on 2017-03-29T13:33:25Z (GMT). No. of bitstreams: 1 Fabiano Neves Nader.pdf: 355299 bytes, checksum: 1b2a2621a49984f5afa2c5660f9b322a (MD5) Previous issue date: 2014-08-19 / This study aims to build the entire theoretical basis for the understanding and use of the Sturm theorem, which is a tool very well grounded to find the number of real roots of a polynomial with real coefficients. During the development of this study sought to illustrate with examples and applications in great detail so that students and teachers are able to understand and use this beautiful algorithm. / Este trabalho tem por finalidade construir toda a base teórica para a compreensão e utilização do Teorema de Sturm, que é uma ferramenta muito bem fundamentada para encontrar a quantidade de raízes reais de um polinômio com coeficientes reais. Durante o desenvolvimento desse estudo procurei ilustrar com exemplos e aplicações de forma bem detalhada para que alunos e professores tenham condições de entender e utilizar este belíssimo algoritmo.

Cette thèse traite du comptage d'extensions cubiques relatives. Dans le premier chapitre on traite un travail commun avec Henri Cohen. Soit k un corps de nombres. On donne une formule asymptotique pour le nombre de classes d'isomorphisme d'extensions cubiques L/k telles que la clôture galoisienne de L/k contienne une extension quadratique fixée K_2/k. L'outil principal est la théorie de Kummer. Dans le second chapitre, on suppose k un corps quadratique imaginaire (avec nombre de classes 1) et on décrit un algorithme pour énumérer toutes les classes d'isomorphisme d'extensions cubiques L/k jusqu'à une certaine borne X sur la norme du discriminant relatif. / This thesis is about counting relative cubic extensions. In the first chapter we describe a joint work with Henri Cohen. Let k be a number field. We give an asymptotic formula for the number of isomorphism classes of cubic extensions L/k such that the Galois closure of L/k contains a fixed quadratic extension K_2/k. The main tool is Kummer theory. In the second chapter, we suppose k to be an imaginary quadratic number field (with class number 1) and we describe an algorithm for listing all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant ideal.