Le sujet de cette thèse est l'étude d'idéaux monomiaux de l'anneau de polynômes S qui ont une résolution linéaire. D'après un résultat remarquable de Bayer et Stilman et en utilisant la polarisation, la classification des idéaux monomiaux ayant une résolution linéaire est équivalente à la classification des idéaux monomiaux libres de carrés ayant une résolution linéaire. Pour cette raison dans cette thèse nous considérons seulement le cas d'idéaux monomiaux libres de carrés. De plus, le théorème de Eagon-Reiner établit une dualité entre les idéaux monomiaux libres de carrés ayant une résolution linéaire et les idéaux monomiaux libres de carrés Cohen-Macaulay, ce qui montre que le problème de classification des idéaux monomiaux libres de carrés ayant une résolution linéaire est très difficile. Nous rappelons que les idéaux monomiaux libres de carrés sont en correspondance biunivoque avec les complexes simpliciaux d'une part, et d'autre part avec les clutters. Ces correspondances nous motivent pour utiliser les propriétés combinatoires des complexes simpliciaux et des clutters pour obtenir des résultats algébriques. La classification des idéaux monomiaux libres de carrés ayant une résolution linéaire engendrés en degré 2 a été faite par Froberg en 1990. Froberg a observé que l'idéal des circuits d'un graphe G a une résolution 2-linéaire si et seulement si G est un graphe de cordes, i.e. il n'a pas de cycles minimaux de longueur plus grande que 4. Dans [Em, ThVt, VtV, W] les auteurs ont partiellement généralisé les résultats de Froberg à des idéaux engendrés en degré >2. Ils ont introduit plusieurs définitions de clutters de cordes et démontré que les idéaux de circuits correspondant ont une résolution linéaire. Nous pouvons voir les cycles du point de vue topologique, comme la triangulation d'une courbe fermée, dans cette thèse nous utiliserons cette idée pour étudier des clutters associés à des triangulation de pseudo-manifolds en vue d'obtenir une généralisation partielle des résultats de Froberg à des idéaux engendrés en degré >2. Nous comparons notre travail à ceux de [Em, ThVt, VtV, W]. Nous présentons nos résultats dans le chapitres 4 et 5. / In this thesis, we study square-free monomial ideals of the polynomial ring S which have a linear resolution. By remarkable result of Bayer and Stilman [BS] and the technique of polarization, classification of ideals with linear resolution is equivalent to classification of square-free monomial ideals with linear resolution. For this reason, we consider only square-free monomial ideals in S. However, classification of square-free monomial ideals with linear resolution seems to be so difficult because by Eagon-Reiner Theorem [ER], this is equivalent to classification of Cohen-Macaulay ideals. It is worth to note that, square-free monomial ideals in S are in one-to-one correspondence to Stanley-Reisener ideals of simplicial complexes on one hand and the circuit ideal of clutters from another hand. This correspondence motivated mathematicians to use the combinatorial and geometrical properties of these objects in order to get the desired algebraic results. Classification of square-free monomial ideals with 2-linear resolution, was successfully done by Froberg [Fr] in 1990. Froberg observed that the circuit ideal of a graph G has a 2-linear resolution if and only if G is chordal, that is, G does not have an induced cycle of length > 3. In [Em, ThVt, VtV, W] the authors have partially generalized the Fr¨oberg's theorem for degree greater than 2. They have introduced several definitions of chordal clutters and proved that, their corresponding circuit ideals have linear resolutions. Viewing cycles as geometrical objects (triangulation of closed curves), in this thesis we try to generalize the concept of cycles to triangulation of pseudo-manifolds and get a partial generalization of Froberg's theorem for higher dimensional hypergraphs. All the results in Chapters 4 and 5 and some results in Chapter 3 are devoted to be the original results.

Submitted by Clebson Anjos (clebson.leandro54@gmail.com) on 2015-05-18T18:12:57Z No. of bitstreams: 1 arquivototal.pdf: 13930019 bytes, checksum: d5e52fb67904848f89fafaf5ec93c06d (MD5) / Approved for entry into archive by Clebson Anjos (clebson.leandro54@gmail.com) on 2015-05-18T18:15:19Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 13930019 bytes, checksum: d5e52fb67904848f89fafaf5ec93c06d (MD5) / Made available in DSpace on 2015-05-18T18:15:19Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 13930019 bytes, checksum: d5e52fb67904848f89fafaf5ec93c06d (MD5) Previous issue date: 2013-09-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation is focused on the Euler's theorem for polyhedra homeomorphic to the sphere. Present statements made by Cauchy, Poincaré and Legendre. As a consequence we show that there are only ve regular convex polyhedra, called polyhedra Plato. / Esta dissertação tem como tema central o Teorema de Euler para poliedros homeomorfos à esfera. Apresentamos demonstrações feitas por Cauchy, Poincaré e Legendre. Como consequência mostramos a existência de apenas cinco poliedros convexos regulares, os chamados poliedros de Platão.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation aims at a detailed study of the Hilbert function and graded Betti number and the statements of some theorems that relate these two theories. We will also a brief overview on free resolutions and minimal simplicial complex to demonstrate the theorem of Bayer, Sturmfels and Peeva and then, we will conclude with the following result: given an ideal J we will display a set X P2 such that the minimal resolution the ideal of de nition of X has the same Betti diagram of the minimal resolution of J. / Esta disserta¸c˜ao tem como objetivo um estudo detalhado da fun¸c˜ao de Hilbert e do n´umero graduado de Betti e as demonstra¸c˜oes de alguns teoremas que relacionam essas duas teorias. Faremos tamb´em um breve apanhado sobre resolu¸c˜oes livres minimais e complexo simplicial para demonstrar o teorema de Bayer, Peeva e Sturmfels e por fim e n˜ao menos importante concluiremos com o seguinte resultado: dado um ideal J exibiremos um conjunto X P2 tal que a resolu¸c˜ao minimal do ideal de defini¸c˜ao de X tenha o mesmo diagrama de Betti da resolu¸c˜ao minimal de J.

La géométrie algébrique réelle est dans sa définition la plus simple, l'étude des ensembles de solutions d'un système d'équations polynomiales à coefficients réelles. Dans cette vaste thématique, on se concentre sur les intersections de quadriques où déjà le cas de trois quadriques reste largement ouvert. Notre sujet peut être résumé comme l'étude topologique des variétés algébriques réelles et l'interaction entre leur topologie d'une part et leur déformations et dégénérations d'autre part, un problème issu du 16ième problème de Hilbert et enrichi par des développements récents. Au cours de cette thèse, nous allons nous focaliser sur les intersections maximales de quadriques réelles et en particulier démonter l'existence de telles intersections en utilisant des développements issus des recherches effectuées depuis la fin des années 80. Dans le cas d'intersections de trois quadriques, nous allons mettre en évidence le lien très étroits entre ces intersections d'une part et les courbes planes d'autre part, et démontrer que l'étude des M-courbes (une des problématiques du 16ième problème de Hilbert) peut se faire à travers l'étude des intersections maximales. Nous utiliserons ensuite les résultats sur les courbes planes nodales afin de déterminer dans certains cas les classes de déformations d'intersections de trois quadriques réelles. / Real algebraic geometry is in its simplest definition, the study of sets of solutions of a system of polynomial equations with real coefficients. In this theme, we focus on the intersections of quadrics where already the case of three quadrics remains wide open. Our subject can be summarized as the topological study of real algebraic varieties and interaction between their topology on the one hand and their deformations and degenerations on the other hand, a problem coming from the 16th Hilbert problem and enriched by recent developments. In this thesis, we will focus on maximum intersections of real quadrics and particularly prove the existence of such intersections using research developments made since the late 80. In the case of intersections of three quadrics, we will point the very close link between the intersections on the one hand and on the other plane curves, and show that the study of M-curves (one of the problems of the 16th Hilbert problem) may be done through the study of maximum intersections. Next, we will use the study on nodal plane curves to determine in some cases deformation classes of intersections of three real quadrics.

En utilisant le concept des fonctions de partition , nous étudions le comportement asymptotique des nombres de Betti gradués des puissances d’idéaux homogènes dans un polynôme sur un corp.Pour un Z-graduer positif, notre résultat principal affirme que les nombres de Betti des puissances est codé par un nombre fini des polynômes. Plus précisément, Z^2 peut être divisé en un nombre fini des régions telles que, dans chacun d’eux, dimk Tor^{S}_{i} (I^t,k)μ est un quasi-polynôme en (μ,t). Ce affine, dans une situation graduée, le résultat de Kodiyalam sur nombres de Betti des puissances dans [33].La déclaration principale traite le cas des produits des puissances d’idéaux homogènes dans un algèbre Z^d -graduée , pour un graduer positif, dans le sens de [37] et il est généralise également pour les filtrations I -good.Dans la deuxième partie, en utilisant la version paramétrique de l’algorithme de Barvinok, nous donnons une formule fermée pour les fonctions de Hilbert non-standard d’anneaux de polynômes, en petites dimensions. / Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. For a positive Z-grading, our main result states that the Betti numbers of powers is encoded by finitely many polynomials. More precisely, Z^2 can be splitted into a finite number of regions such that, in each of them, dim_k Tor^{S}_{i} (I^t,k)μ is a quasi-polynomial in (μ,t). This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers in [33]. The main statement treats the case of a power products of homogeneous ideals in a Z^d -graded algebra, for a positive grading, in the sense of [37] and it is also generalizes to I -good filtrations . In the second part , using the parametric version of Barvinok’s algorithm, we give a closed formula for non-standard Hilbert functions of polynomial rings, in low dimensions.

In this thesis, we will work with the nodal sets of Laplace eigenfunctions on a few simple manifolds, like the sphere and the flat torus. We will obtain bounds on the total Betti number of the nodal set that depend on the corresponding eigenvalue. Our work generalize Courant's theorem. / Dans ce mémoire, nous travaillons sur les ensembles nodaux de combinaisons de fonctions propres du laplacien, particulièrement sur la sphère et le tore plat. On bornera les nombres de Betti de ces ensembles en fonction de la valeur propre maximale. D'une certaine façon, cela généralise le fameux théorème de Courant.

This thesis deals with toric ideals associated with finite simple graphs. In particular we establish some results pertaining to the nature of the generators and syzygies of toric ideals associated with finite simple graphs. The first result dealt with in this thesis expands upon work by Favacchio, Hofscheier, Keiper, and Van Tuyl which states that for G, a graph obtained by "gluing" a graph H1 to a graph H2 along an induced subgraph, we can obtain the toric ideal associated to G from the toric ideals associated to H1 and H2 by taking their sum as ideals in the larger ring and saturating by a particular monomial f. Our contribution is to sharpen the result and show that instead of a saturation by f, we need only examine the colon ideal with f^2. The second result treated by this thesis pertains to graded Betti numbers of toric ideals of complete bipartite graphs. We show that by counting specific subgraphs one can explicitly compute a minimal set of generators for the corresponding toric ideals as well as minimal generating sets for the first two syzygy modules. Additionally we provide formulas for some of the graded Betti numbers. The final topic treated pertains to a relationship between the fundamental group the finite simple graph G and the associated toric ideal to G. It was shown by Villareal as well as Hibi and Ohsugi that the generators of a toric ideal associated to a finite simple graph correspond to the closed even walks of the graph G, thus linking algebraic properties to combinatorial ones. Therefore it is a natural question whether there is a relationship between the toric ideal associated to the graph G and the fundamental group of the graph G. We show, under the assumption that G is a bipartite graph with some additional assumptions, one can conceive of the set of binomials in the toric ideal with coprime terms, B(IG), as a group with an appropriately chosen operation ⋆ and establish a group isomorphism (B(IG), ⋆) ∼= π1(G)/H where H is a normal subgroup. We exploit this relationship further to obtain information about the generators of IG as well as bounds on the Betti numbers. We are also able to characterise all regular sequences and hence compute the depth of the toric ideal of G. We also use the framework to prove that IG = (⟨G⟩ : (e1 · · · em)^∞) where G is a set of binomials which correspond to a generating set of π1(G). / Thesis / Doctor of Philosophy (PhD)

Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.

<p>This thesis is concerned with a structure called the Reeb graph. There are three main problems considered. The first is devising an efficient algorithm for comnstructing the Reeb graph of a simplicial complex with respect to a generic simplex-wise linear real-valued function. We present an algorithm that builds the Reeb graph in almost optimal worst-case deterministic time. This was the first deterministic algorithm with the time bound which is linear up to a logarithmic factor. Without loss of generality, the complex is assumed to be 2-dimensional. The algorithm works by sweeping the function values and maintaining a spanning forest for the preimage, or the level-set, of the value. Using the observation that the operations that change the level-sets are off-line, we was able to achieve the above bound.</p><p>The second topic is the dynamic Reeb graphs. As the function changes its values, the Reeb graph also changes. We are interested in updating the Reeb graph. We reduce this problem to a graph problem that we call retroactive graph connectivity. We obtain algorithms for dynamic Reeb graphs, by using data structures that solve the graph problem. </p><p>The third topic is an argument regarding the complexity of computing Betti numbers. This problem is also related to the Reeb graph by means of the vertical Homology classes. The problem considered here is whether the realization of a simplicial complex in the Euclidean 4-space can result in an algorithm for computing its Homology groups faster than the usual matrix reduction methods. Using the observation that the vertical Betti numbers can always be computed more efficiently using the Reeb graph, we show that the answer to this question is in general negative. For instance, given a square matrix over the field with two elements, we construct a simplicial complex in linear time, realized in euclidean 4-space and a function on it, such that its Horizontal homology group, over the same field, is isomorphic with the null-space of the matrix. It follows that the Betti number computation for complexes realized in the 4-space is as hard as computing the rank for a sparse matrix.</p> / Dissertation

A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete measures such as the degree of polynomials, and complexity of Pfaffian functions. A generalisation of the Thom-Milnor Bound [17, 19] for sets defined by the conjunction of equations and non-strict inequalities is presented, in the new context of sets definable in o-minimal structures using the axiomatic complexity measure. Next bounds are produced for sets defined by Boolean combinations of equations and inequalities, through firstly considering sets defined by sign conditions, then using this to produce results for closed sets, and then making use of a construction to approximate any set defined by a Boolean combination of equations and inequalities by a closed set. Lastly, existing results [12] for sets defined using quantifiers on an open or closed set are generalised, using a construction from Gabrielov and Vorobjov [11] to approximate any set by a compact set. This results in a method to find a general bound for any set definable in an o-minimal structure in terms of the axiomatic complexity measure. As a consequence for the first time an upper bound for sub-Pfaffian sets defined by arbitrary formulae with quantifiers is given. This bound is singly exponential if the number of quantifier alternations is fixed.