Orientador: Paulo Roberto Brumatti / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-08-10T20:04:54Z (GMT). No. of bitstreams: 1 Bianchi_AngeloCalil_M.pdf: 609753 bytes, checksum: d05b2d15e03b1e36b83018ed28d8da63 (MD5) Previous issue date: 2008 / Resumo: O objetivo geral desse trabalho é entender a teoria das bases SAGBI num ponto de vista estrutural, buscando critérios para sua existência e resultados que comprovem sua eficácia para o estudo de certas k-subalgebras de k[x], bem como estudar a teoria geral das derivações sobre anéis de polinômios, suas localizações e quocientes, visando explorar as propriedades algébricas do núcleo destas derivações e as estruturas das k-subalgebras de k[x] que podem ser vistas como tais núcleos. O objetivo específico é estudar a teoria algébrico-geométrica para k-derivações em k[x], desenvolvida por Shigeru Kuroda, e utilizar dessa teoria para estabelecer uma condição para que o núcleo de uma tal derivação seja uma k-subalgebra finitamente gerada e outra para que este possua uma base SAGBI finita. Em cada momento ao longo do trabalho também é desejado enfatizar o comportamento das k-derivações que são localmente nilpotentes e obter uma forma algorítmica para determinar os geradores de seus núcleos, no caso particular da derivação ao possuir uma slice / Abstract: The general objective of this work is to understand the SAGBI bases theory from a structural point of view, seeking criterias for it¿s existence and results that prove it¿s effitiency in the study of certain subalgebras of k[x], as well as to study the general theory of derivations over polynomial rings, it¿s localizations and quotients, in order to explore the algebraic properties of the kernel of this derivations and the structures of the k-subalgebras of k[x] that may be seen as such kernels. The specific objective is to study the algebraic-geometric theory of k-derivations in k[x], developed by Shigeru Kuroda, and to use this theory to stabilish a condition for the kernel of one such derivation to be a finitely generated k-subalgebra and another condition for this derivation to have finite SAGBI base. Along this work we also want to emphasize the behavior of locally nilpotent k-derivations and to obtain an algorithmic way to determine the generators of it¿s kernels, in the particular case that the derivation has a slice / Mestrado / Matematica / Mestre em Matemática

Nesta tese estudamos possíveis extensões do modelo CPN-1 em (2+1) dimensões. Provamos que quando tomado na representação fundamental à esquerda ele é renormalizável e não possui divergências infravermelhas perigosas. O mesmo não ocorre se o campo principal . Mostramos que a inclusão de férmions, minimamente acoplados ao campo de calibre, traz alguma melhoria no comportamento das divergências infravermelhas no setor de calibre em ordem dominante em 1/N. Discutimos também a invariância de calibre no procedimento de renormalização. / In this thesis investigate possible extensions of the (2+1) dimensional CPN-1 model to the noncommutative space. Up to leading nontrivial order of 1/N, we prove that the model restricted to the left fundamental representation is renormalizable and does not have dangerous infrared divergences. By contrast, IF the pricipal Field transforms in accord with the adjoint representation, linearly divergent, nonintegrable singularities are present in the two point function of the auxiliary gauge Field and also in the leading correction to the self-energy of the Field. It is showed that the inclusion of fermionic matter, minimally coupled to the gauge Field, ameliorates this behavior by eliminating infrared divergences in the gauge sector at the leading 1/N order. Gauge invariance of the renormalization is also discussed.

Orientadores: Fernando Eduardo Torres Orihuela, Cícero Fernandes de Carvalho / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T03:11:40Z (GMT). No. of bitstreams: 1 Peixoto_Rafael_D.pdf: 876847 bytes, checksum: ae0f5d0ea0f2c3e3d550bc60eb1ac66a (MD5) Previous issue date: 2011 / Resumo: Definidas sobre uma F-álgebra, os conceitos de função peso e função peso fraco foram introduzidos de forma a simplificar a teoria dos códigos corretores de erros que utilizam ferramentas da geometria algébrica. Porém, todos os códigos suportados por estes conceitos estão intimamente ligados à códigos provenientes de curvas algébricas, ou seja, os códigos geométricos de Goppa. Uma modificação da noção de função peso foi apresentada permitindo assim construir códigos lineares sobre variedades algébricas. Nesta tese, apresentamos uma generalização da teoria de funções pesos fracos que possibilitou a construção de códigos sobre variedades de dimensão arbitrária. Determinamos uma cota para a distância mínima destes códigos, e finalmente, apresentamos uma caracterização tanto para as álgebras munidas de funções pesos quanto para as álgebras munidas de um conjunto especial de funções pesos fracos / Abstract: Defined on a F-algebra, the concepts of weight and near weight function were introduced to simplify the theory of error correcting codes using tools from algebraic geometry. However, all codes supported by these theories are geometric Goppa codes. The concept of weight function was generalized and used to construct linear codes on algebraic varieties. In this thesis, we present a generalization of near weights theory able to construct codes on higher dimensional varieties, and we define a formula for the minimum distance of such codes. Finally, we characterize the algebras with a weight function and the algebras admitting a special set of two near weight functions / Doutorado / Matematica / Doutor em Matemática

The thrust of this thesis is to describe base change BC_E/F at the level of chamber homology and K-theory for some p-adic groups, such as SL(2,F), GL(1,F) and GL(2,F). Here F is a non-archimedean local field and E is a Galois extension of F. We have had to master the representation theory of SL(2) and GL(2) including the Langlands parameters. The main result is an explicit computation of the effect of base change on the chamber homology groups, each of which is constructed from cycles. This will have an important connection with the Baum-Connes correspondence for such p-adic groups. This thesis involved the arithmetic of fields such as E and F, geometry of trees, the homology groups and the Weil group W_F.

Orientador: Paulo Roberto Brumatti / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:10:19Z (GMT). No. of bitstreams: 1 Veloso_MarceloOliveira_M.pdf: 605567 bytes, checksum: 5882cdcae8b9c04096f915755c89a683 (MD5) Previous issue date: 2005 / Resumo: O objetivo principal deste trabalho é o estudo do número de soluções de equações polinomiais definidas sobre corpos finitos. Para isto utilizamos resultados básicos sobre a soma de Caracteres e resultados sobre o número de soluções de uma Forma Quadrática. Na nossa abordagem procuramos utilizar técnicas bem elementares, apesar disto implicar num número maior de cálculos. Contudo este método permitiu estudar e determinar fórmulas para o número de soluções de determinadas equações polinomiais muito estudadas, sem a necessidade de ferramentas mais elaboradas. Dentre as aplicações das fórmulas obtidas, temos alguns exemplos de curvas algébricas planas cujo número de pontos racionais atingem a cota de Weil, ou seja, curvas maximais que são de grande interesse em teoria dos códigos. Também conseguimos exemplos de variedades projetivas sobre corpos finitos cujo número de pontos atingem a cota de Weil-Deligne / Abstract: The main objective of this work is to study the number of solutions of polynomial equations over finite fields. For that we used basic results on Character sums and on the number of solutions of a Quadratic Form. This approach uses elementary techniques even considering the increasing on computations. Therefore this method allowed us to study and determine formulae for the number of solutions of certain polynomial equations well known, without the need of more sophisticated tools. Among the applications of the obtained formulae, we have some examples of plane algebraic curves which number of rational points achieve the Weil bound, that is, maximal curves which are of great interest in code theory. In addition, other examples were obtained of projective manifolds over finite fields which number of points achieve the Weil-Deligne bound / Mestrado / Algebra / Mestre em Matemática

No description available.

Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties. Let R be the polynomial ring Ζ[x1,…,xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal I_h generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for Jh, we identify a basis for the quotient R/Jh. We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h>h', we have Ih ⊂ Ih' and Jh ⊂ Jh'. We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety. The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator X and a Hessenberg function h. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as h varies, the quotient R/Jh may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.

De nombreuses approches à la gravité quantique suggèrent que la description usuelle de l’espace-temps ne serait pas adaptée à la description des phénomènes physiques impliquant à la fois des processus gravitationnels et quantiques. Une meilleure description pourrait consister à munir l’espace-temps d’une structure non-commutative en remplaçant les coordonnées locales sur la variété par des opérateurs ne commutant pas deux-à-deux. Il s’ensuit que le comportement des théories de champs construites sur de tels espaces diffère en général de celui des théories de champs ordinaires. L’étude de ces possibles nouvelles propriétés est l’objet de la théorie non-commutative des champs (TNCC) dont nous étudions certains des aspects.Dans le présent mémoire, nous considérons deux familles d’espaces quantiques dont l’algèbres de coordonnées admet une structure d’algèbre de Lie. La première famille est caractérisée par l’algèbre su(2) et apparait dans le cadre de modèle de gravité quantique en 3 dimensions, ainsi que dans certains modèles de « brane » et de « group field theory ». La seconde famille d’espaces quantiques est connue sous le nom de kappa-Minkowski. L’intérêt de cet espace réside dans le fait qu’il est défini comme l’espace homogène associé à l’algèbre de Hopf de kappa-Poincaré. Cette dernière définit une déformation, à l’échelle de Planck, de l’algèbre de Poincaré et s’avère être étroitement liée à certains modèles de gravité quantique.Afin d’étudier les TNCC, il est commode de représenter l’espace quantique comme une algèbre non-commutative de fonctions munie d’un produit déformé appelé « star-product ». Une façon canonique de construire un tel produit consiste à se servir d’outils d’analyse harmonique et à adapter le schéma de quantification de Weyl (originellement introduit dans le cadre de la mécanique quantique) à l’algèbre considérée. Les expressions de star-product associé aux espaces susmentionnés sont dérivées de manière explicite. Nous montrons en particulier que des familles de star-product inéquivalents peuvent être classifiées par des considérations cohomologiques. Nous étudions enfin les propriétés quantiques de différents modèles de TNCC scalaire quartique construits à l’aide de ces star-product. Dans le cas où l’espace quantique est caractérisé par l’algèbre su(2), nous trouvons que la fonction 2-point est fini à l’ordre une boucle, le paramètre de déformation jouant le rôle d’une coupure ultraviolette et infrarouge. Dans le cas de kappa-Minkowski, nous insistons sur l’invariance sous kappa-Poincaré de l’action fonctionnelle et montrons que certains modèles de TNCC scalaire quartique divergent moins que dans le cas commutatif. Par ailleurs, la fonction 4-point est trouvée finie à l’ordre une boucle. Nos résultats, ainsi que leurs conséquences, sont finalement discutés. / As many theoretical studies point out, the classical description of spacetime, as a continuum, might be no longer adequate to reconcile gravity with quantum mechanics at very high energy (the relevant energy scale being often regarded as the Planck scale). Instead, a more appropriate description could be provided by the data of a noncommutative algebra of coordinate operators replacing the usual commutative local coordinates on smooth manifold. Once the noncommutative nature of spacetime is assumed, it is to expect that the (classical and quantum) properties of field theories on noncommutative background differ from the ones of field theories on classical background. This is the aim of Non-Commutative Field Theory (NCFT) to explore and study these new properties.In the present dissertation, we consider two families of quantum spacetimes of Lie algebra type noncommutativity. The first family is characterised by su(2) noncommutativity and appears in the description of some models of quantum gravity in 3-dimensions. The other family of quantum spacetimes is known in the physics literature as the 4-d kappa-Minkowski space. The importance of this quantum spacetime lies into the fact that its symmetries are provided by the (quantum) kappa-Poincaré algebra (a deformation of the classical Poincaré algebra) together with the fact that the deformation parameter 'kappa', which is of mass dimension, provides a natural energy scale at which the quantum gravity effects may be relevant (and is often regarded as being related to the Planck scale). For these reasons, the kappa-Minkowski space appears as a good candidate for a spacetime to be involved in the description of Doubly Special Relativity and Relative Locality models.To study NCFT it is often convenient to introduce a star product characterising the (noncommutative) C*-algebra of fields modelling the quantum spacetime under consideration. We emphasise that a canonical star product can be obtained by using the group algebraic structures underlying the construction of such Lie algebra type quantum spaces, namely by making use of harmonic analysis on the corresponding Lie group together with the Weyl quantisation scheme. The explicit derivation of such star product for kappa-Minkowski is given. In addition, we show that su(2) Lie algebras of coordinate operators related to quantum spaces with su(2) noncommutativity can be conveniently represented by SO(3)-equivariant poly-differential involutive representations and show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). We finally indicate a convenient way to extend this construction to other semi-simple but non simply connected Lie groups by making use of results from group cohomology with value in an abelian group that would replace the constraints stemming from the simple Wigner theorem.Then, we investigate the quantum properties of various models of interacting scalar field theory on noncommutative background making use of the aforementioned star product formalism to construct physically reasonable expressions for the action functional. Considering quantum spacetime with su(2) noncommutativity, we find that the one-loop 2-point function for complex scalar field theories with quartic interactions is finite, the deformation parameter playing the role of a natural UV cut-off. Special attention is paid to the derivation of the one-loop corrections to both the 2-point and 4-point functions for various models of kappa-Poincaré invariant scalar field theory with quartic interactions. In that case, we show that for some models the 2-point function divergences linearly thus slightly milder than their commutative counterpart, while the one-loop 4-point function is shown to be finite. The results we obtained together with their consequences are finally discussed.

In the need of computational tools for epsilon-multiplicity, we provide a criterion for a module with a rank E inside a free module F to have rational epsilon-multiplicity in terms of the finite generation of the saturation Rees algebra of E. In this case, the multiplicity can be related to a Hilbert multiplicity of certain graded algebra. A particular example of this situation is provided: it is shown that the epsilon-multiplicity of monomial modules is Noetherian. Numerical evidence is provided that leads to a conjecture formula for the epsilon-multiplicity of certain monomial curves in the 3-affine space.

<p>Let S be a polynomial ring K[x1,...,xn] over a field K and let F be a non-negatively graded free module over S generated by m basis elements. In this thesis, we study four kinds of upper bounds: degree bounds for Gröbner bases of submodules of F, bounds for arithmetic degrees of S-ideals, regularity bounds for radicals of S-ideals, and Stillman bounds. </p> <p><br></p> <p>Let M be a submodule of F generated by elements with degrees bounded above by D and dim(F/M)=r. We prove that if M is graded, the degree of the reduced Gröbner basis of M for any term order is bounded above by 2[1/2((Dm)^{n-r}m+D)]^{2^{r-1}}. If M is not graded, the bound is 2[1/2((Dm)^{(n-r)^2}m+D)]^{2^{r}}. This is a generalization of bounds for ideals in a polynomial ring due to Dubé (1990) and Mayr-Ritscher (2013).</p> <p><br></p> <p>Our next results are concerned with a homogeneous ideal I in S generated by forms of degree at most d with dim(S/I)=r. In Chapter 4, we show how to derive from a result of Hoa (2008) an upper bound for the regularity of sqrt{I}, which denotes the radical of I. More specifically we show that reg(sqrt{I})<= d^{(n-1)2^{r-1}}. In Chapter 5, we show that the i-th arithmetic degree of I is bounded above by 2*d^{2^{n-i-1}}. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type.</p> <p><br></p> <p>In the last chapter, we explain our progress in attempting to make Stillman bounds explicit. Ananyan and Hochster (2020) were the first to show the existence of Stillman bounds. Together with G. Caviglia, we observe that a possible way of making their results explicit is to find an effective bound for an invariant called D(k,d) and supplement it into their proof. Although we are able to obtain this bound D(k,d) and realize Stillman bounds via an algorithm, it turns out that the computational complexity of Ananyan and Hochster's inductive proof would make the bounds too large to be meaningful. We explain the bad behavior of these Stillman bounds by giving estimates up to degree 3.</p>