In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence. In the second chapter we introduce the class number, proper class number and complete class number as well as two refinements, which facilitate the development of a connection with binary quadratic forms. Our third chapter is devoted to deriving several class number formulas in terms of divisors of the determinant. This chapter also contains lower bounds on the class number for bilinear forms and classifies when these bounds are attained. Lastly, we use the class number formulas to rigorously develop Kronecker's connection between binary bilinear forms and binary quadratic forms. We supply purely arithmetic proofs of five results stated but not proven in the original paper. We conclude by giving an application of this material to the number of representations of an integer as a sum of three squares and show the resulting formula is equivalent to the well-known result due to Gauss.

Les calculs numériques ont été effectués à l'aide du logiciel SAGE. / En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers.

We consider systems of polynomial equations and inequalities to be solved in integers. By applying the circle method, when the number of variables is large and the system is geometrically well-behaved we give an asymptotic estimate for the number of solutions of bounded size. In the case of R homogeneous equations having the same degree d, a classic theorem of Birch provides such an estimate provided the number of variables is R(R+1)(d-1)2^d-1+R or greater and the system is nonsingular. In many cases this conclusion has been improved, but except in the case of diagonal equations the number of variables needed has always grown quadratically in R. We give a result requiring only d2^dR+R variables, obtaining linear growth in R. When d = 2 or 3 we require only that the system be nonsingular; when d<4 we require that the coefficients of the equations belong to a certain explicit Zariski open set. These conditions are satisfied for typical systems of equations, and can in principle be checked algorithmically for any particular system. We also give an asymptotic estimate for the number of solutions to R polynomial inequalities of degree d with real coefficients, in the same number of variables and satisfying the same geometric conditions as in our work on equations. Previously one needed the number of variables to grow super-exponentially in the degree d in order to show that a nontrivial solution exists.

En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers. / Les calculs numériques ont été effectués à l'aide du logiciel SAGE.

Made available in DSpace on 2016-06-02T20:28:29Z (GMT). No. of bitstreams: 1 5894.pdf: 474601 bytes, checksum: 5d3ab33b83cca94abae2ce5efc49bf32 (MD5) Previous issue date: 2014-03-21 / Financiadora de Estudos e Projetos / Let Ω be a deformed tube in(continue...) / Seja um tubo deformado em (continua)

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Seja ℓ > 3 um primo Ãmpar. Sejam So, S+, S_ conjuntos finitos mutuamente disjuntos de primos racionais. Para qualquer nÃmero real suficientemente grande X > 0, baseando-nos em [16], damos neste trabalho, um limite inferior do nÃmero de corpos quadrÃticos imaginÃrios k que satisfazem as seguintes condiÃÃes: o discriminante de k à maior que -X o nÃmero de classe de k à nÃo divisÃvel por ℓ, todo q â So se ramifica, todo q â S+ se decompÃe e todo q â S_ à inerte em k, respectivamente. / Let ℓ > 3 be an odd prime. Let So, S+, S_ be mutually disjoint finite sets of rational primes. For any suficiently large real number X > 0, basing ourselves on [16], we give this paper a lower bound of the number of imaginary quadratic fields k which satisfy the following conditions: the discriminant of k is greater than -X, the class number ok is not divisible by ℓ, every q â So ramifies, every q â S+ splits and every q â S_ is inert in k, respectively.

Neste trabalho procuramos realizar uma apresentação autocontida sobre os conceitos da teoria algébrica de formas quadráticas e sobre os anéis graduados que surgiram no desenvolvimento desta teoria. Iniciamos procurando esclarecer o sentido da equivalência entre as várias acepções do conceito de forma quadrática. Após a apresentação de ingredientes e resultados geométricos, fazemos um extrato da teoria dos anéis de Witt, conceito que originou a moderna teoria algébrica de formas quadráticas. Disponibilizamos os elementos fundamentais para a formulação das teorias de cohomologia, nos concentrado no desenvolvimento da teoria de cohomologia profinita e, sobretudo, galoisiana. Descrevemos os funtores K0, K1 e K2 da K-teoria clássica e também a K-teoria de Milnor, que é mais adequada para formular questões sobre formas quadráticas. Finalizamos o trabalho com a apresentação de alguns conceitos da Teoria dos Grupos Especiais, uma codificação em primeira-ordem da teoria algébrica das formas quadráticas e exemplificamos sua importância, fornecendo um extrato da prova realizada por Dickmann-Miraglia da conjectura de Marshall sobre assinaturas, que se baseia fortemente nesta teoria. / In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of \"equivalence\"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K0, K1 and K2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshalls conjecture on signatures, which relies heavily on this theory.

No description available.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-10-22T13:10:57Z No. of bitstreams: 2 Disertação - Luiz Fernando Rodrigues Santana - 2018.pdf: 1262248 bytes, checksum: 28c77ae261289cc58c11db648cd4572b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-10-22T13:16:35Z (GMT) No. of bitstreams: 2 Disertação - Luiz Fernando Rodrigues Santana - 2018.pdf: 1262248 bytes, checksum: 28c77ae261289cc58c11db648cd4572b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-10-22T13:16:35Z (GMT). No. of bitstreams: 2 Disertação - Luiz Fernando Rodrigues Santana - 2018.pdf: 1262248 bytes, checksum: 28c77ae261289cc58c11db648cd4572b (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-10-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This text presents the properties and definitions of p-adic numbers linked to the definition of quadratic forms. Hasse's theorem: “Every quadratic form, with 5 variables or more, has non-trivial p-adic zeros” exemplifies the Local- Global Principle, which in turn ensures that if a polynomial equation has non-trivial rational zeros if, and only if, It has non-trivial zeros over R and about Qp, p prime. / Este texto apresenta as propriedades e as definições de números p-ádicos atreladas à definição de formas quadráticas. O teorema de Hasse: “Toda forma quadrática, com 5 variáveis ou mais, possui zeros p-ádicos não triviais” exemplifia o Princípio Local Global, que por sua vez garante que se uma equação polinomial possui zeros racionais não triviais se, e somente se, possui zeros não triviais sobre R e sobre Qp, p primo.

Orientador: Edson Moschim / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-04T21:57:49Z (GMT). No. of bitstreams: 1 DomingosNeto_Adriano_D.pdf: 1544823 bytes, checksum: f09f4717b59d1cce526b7c8746e53efb (MD5) Previous issue date: 2005 / Resumo: Nesta tese, propõe se três novas construções de códigos ortogonais ópticos (OOC), do tipo congruentes, tendo como base a estrutura algébrica do grupo multiplicativo do corpo de Galois GF(p), para aplicação em sistemas de comunicação utilizando a técnica de acesso múltiplo por divisão de códigos ópticos (OCDMA). Os códigos ópticos primos e códigos quadráticos são, pela primeira vez na literatura, gerados a partir de códigos de Slepian (códigos esféricos) e, códigos de resíduos quadráticos, respectivamente. Através do algoritmo da d-cadeia fechada, são obtidos os códigos de primos, como caso particular dos códigos de Slepian. Os códigos quadráticos ópticos são representados por números inteiros quadráticos binários na forma de equações de Diofanto com duas variáveis, de modo que, o reticulado Z2 ou reticulado Â2 fornecem as palavra do código quadrático. O desempenho dos códigos é avaliado usando o critério da probabilidade de erro para situações em que o receptor óptico incorpora um limitador óptico e um fotodiodo APD. O desempenho do sistema é obtido considerando os efeitos da interferência de acesso múltiplo, o ruído balístico do fotodiodo e o ruído térmico do receptor. O desempenho dos códigos propostos é comparado ao desempenho de códigos amplamente divulgados em literatura técnica. Mostra-se ainda que os códigos propostos apresentam desempenho semelhante aos códigos divulgados, tendo como vantagem uma estrutura algébrica de simples implementação e melhor sincronismo / Abstract: This thesis presents a study of optical orthogonal codes (OOe) for application in communication systems using the technique of fiber-optics code division multiple access (OCDMA). The Prime Sequence codes and Quadratic codes are, for the first time in literature, characterized as Slepian group codes (spherical codes) and Quadratic Residues codes, respectively. Through the algorithm of the closed d-chain the Prime Sequence codes are obtained, as a particular case of the Slepian codes. The Quadratic codes are represented by binary quadratic integers in the form of Diophantine equations with two variables, so that, Z2 lattice or Â3 lattice supplies the codeword of the quadratic code. Furthermore, this thesis presents three new constructions of optical orthogonal codes (OOC), construed via congruences having as base the algebraic structure of the multiplicative group of the GaloisField GF(p). The performance of the codes is evaluated using the criterion of the error probability, for situations where the optic receiver incorporates a fiber-optic limiter and a APD photodiode. The performance of the system is evaluated considering the effect of the interference of multiple access, the ballistic noise of the photodiode and the thermal noise of the receiver. The performance of the considered codes is compared with the performance of other codes found in the technical literature. It is observed that the codes considered in this thesis, in this thesis, present similar performance to the reported codes, having as advantage an algebraic structure of simple implementation and better synchronism / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica