<p>The goal of this dissertation is to broaden the classes of ideals for which the generators of residual intersections are known. This is split into two main parts.</p> <p>The first part is Chapter 5, where we prove that, for an ideal I in a local Cohen-Macaulay ring R, under suitable technical assumptions, we are able to express s-residual intersections, for s ≥ μ(I) − 2, in terms of (μ(I) − 2)-residual intersections. This result implies that s- residual intersections can be expressed in terms of links, if μ(I) ≤ ht(I) + 3 and some other hypotheses are satisfied. In Chapter 5, we prove our result using two different methods and two different sets of technical assumptions on the depth conditions satisfied by the ideal I. For Section 5.2 and Section 5.3 we use the properties of Fitting ideals and methods developed in [33] to prove our main result. In these sections, we require I to satisfy the Gs condition and be weakly (s − 2)-residually S2. In Section 5.4, we prove analogous results to those in Section 5.2 and Section 5.3 using disguised residual intersections, a notion developed by Bouca and Hassansadeh in [5].</p> <p>The second part is Chapter 6 where we prove that the n-residual intersections of ideals generated by maximal minors of a 2 × n generic matrix for n ≥ 4 are sums of links. To prove this, we require a series of technical results. We begin by proving the main theorem for this chapter in a special case, using the results of Section 6.1 to compute the generators of the relevant links in a our special case, and then using these generators to compute the Gro ̈bner Basis for the sum of links in Section 6.2. The computation of the Gro ̈bner basis, as well as an application of graph theoretic results about binomial edge ideals [17], allow us to show that our main theorem holds in this special case. Lastly, we conclude our proof in Section 6.3, where we show that n-residual intersections of ideals generated by maximal minors of 2 × n generic matrices commute with specialization maps, and use this to show that the generic n-residual intersections of ideals generated by maximal minors of a 2 × n generic matrix for n ≥ 4 are sums of links. This allows us to prove the main theorem of Chapter 6.</p>

No description available.

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2007. / Includes bibliographical references.

<p>The defining equations of Rees algebras provide a natural pathway to study these rings. However, information regarding these equations is often elusive and enigmatic. In this dissertation we study Rees algebras of particular classes of ideals and modules over hypersurface rings. We extend known results regarding Rees algebras of ideals and modules to this setting and explore the properties of these rings.</p> <p><br></p> <p>The majority of this thesis is spent studying Rees algebras of ideals in hypersurface rings, beginning with perfect ideals of grade two. After introducing certain constructions, we arrive in a setting similar to the one encountered by Boswell and Mukundan in [3]. We establish a similarity between Rees algebras of ideals with linear presentation in hypersurface rings and Rees algebras of ideals with <em>almost</em> linear presentation in polynomial rings. Hence we adapt the methods developed by Boswell and Mukundan in [3] to our setting and follow a path parallel to theirs. We introduce a recursive algorithm of <em>modified Jacobian dual iterations</em> which produces a minimal generating set for the defining ideal of the Rees algebra.</p> <p><br></p> <p>Once success has been achieved for perfect ideals of grade two, we consider perfect Gorenstein ideals of grade three in hypersurface rings and their Rees algebras. We follow a path similar to the one taken for the previous class of ideals. A recursive algorithm of <em>gcd-iterations</em> is introduced and it is shown that this method produces a minimal generating set of the defining ideal of the Rees algebra. </p> <p><br></p> <p>Lastly, we extend our techniques regarding Rees algebras of ideals to Rees algebras of modules. Using <em>generic Bourbaki ideals</em> we study Rees algebras of modules with projective dimension one over hypersurface rings. For such a module $E$, we show that there exists a generic Bourbaki ideal $I$, with respect to $E$, which is perfect of grade two in a hypersurface ring. We then adapt the techniques used by Costantini in [9] to our setting in order to relate the defining ideal of $\mathcal{R}(E)$ to the defining ideal of $\mathcal{R}(I)$, which is known from the earlier work mentioned above.</p> <p><br></p> <p>In all three situations above, once the defining equations have been determined, we investigate certain properties of the Rees algebra. The depth, Cohen-Macaulayness, relation type, and Castelnuovo-Mumford regularity of these rings are explored.</p>

Cette thèse est consacrée à l’étude combinatoire, algébrique et homologique des hyperarbres et des partitions semi-pointées. Nous étudions plus précisément des structures algébriques et homologiques construites à partir des hyperarbres, puis des partitions semi-pointées.Après un bref rappel des notions utilisées, nous utilisons la théorie des espèces de structure afin de déterminer l’action du groupe symétrique sur l’homologie du poset des hyperarbres. Cette action s’identifie à l’action du groupe symétrique liée à la structure anti-cyclique de l’opérade PreLie. Nous raffinons ensuite nos calculs sur une graduation de l’homologie, appelée homologie de Whitney. Cette étude motive l'introduction de la notion d’hyperarbre aux arêtes décorées par une espèce. Une bijection des hyperarbres décorés avec des arbres en boîtes et des partitions décorées permet d’obtenir une formule close pour leur cardinal, à l’aide d’un codage de Prüfer. Nous adaptons ensuite les méthodes de calcul de caractères sur les algèbres de Hopf d’incidence, introduites par W. Schmitt dans le cas de familles de posets bornés, à des familles de posets non bornés vérifiant certaines propriétés. Nous appliquons ensuite cette adaptation aux posets des hyperarbres. Enfin, au cours de notre étude une généralisation des posets des partitions et des posets des partitions pointées apparaît : les poset des partitions semi-pointées. Nous montrons que ces posets sont aussi Cohen-Macaulay, avant de déterminer à l’aide de la théorie des espèces une formule close pour la dimension de l’unique groupe d’homologie non trivial de ces posets / This thesis is dedicated to the combinatorial, algebraic and homological study of hypertrees and semi-pointed partitions. More precisely, we study algebraic and homological structures built from hypertrees and semi-pointed partitions. After recalling briefly the notions needed, we use the theory of species of structures to compute the action of the symmetric group on the homology of the hypertree posets. This action is the same as the action of the symmetric group linked with the anticyclic structure of the PreLie operad. We refine our computations on a grading of the homology : Whitney homology. This study is a motivation for the introduction of the notion of edge-decorated hypertrees. A one-to-one correspondence of decorated hypertrees with box trees and decorated partitions enables us to compute a close formula for the cardinality of decorated hypertrees, thanks to a Prüfer code. Moreover, we adapt computation methods of characters on incidence Hopf algebras, introduced by W. Schmitt for families of bounded posets, to families of unbounded posets satisfying some additional properties, called triangle and diamond posets. We apply these results to the hypertree posets. Finally, we unveil a new family of posets : the semi-pointed partition posets, which generalize both partition posets and pointed partition posets. We show the Cohen-Macaulayness of these posets and obtain, thanks to species theory, a closed formula for the dimension of its unique homology group, which extend the ones established for partition posets and pointed partition posets

Une T-variété est une variété algébrique munie d'une action effective d'un tore algébrique T. Cette thèse est consacrée à l'étude de deux aspects des T-variétés normales affines : les actions du groupe additif et la caractérisation des singularités. Soit X = Spec A une T-variété affine normale et soit D une dérivation homogène localement nilpotente de l'algèbre affine intègre Z^n-graduée A, alors D engendre une action du groupe additif dans X. On donne une classification complète des couples (X, D) dans trois cas : pour les variétés toriques, dans le cas de complexité un, et dans le cas où D est de type fibre. Comme application, on calcule l'invariant de Makar-Limanov (ML) homogène de ces variétés. On en déduit que toute variété d'invariant de ML trivial est birationnelle à Y × P^2 , pour une certaine variété Y . Inversement, pour toute variété Y , il existe une T-variété affine X d'invariant de ML trivial birationnelle a Y × P2. Dans la seconde partie concernant les singularités d'une T-variété X, on calcule les images directes supérieures du faisceau structural d'une désingularisation de X. Comme conséquence, on donne un critère pour qu'une T-variété ait des singularités rationnelles. On présente aussi une condition pour qu'une T-variété soit de Cohen-Macaulay. Comme application, on caractérise les singularités elliptiques des surfaces quasi-homogènes.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-31T12:43:01Z No. of bitstreams: 1 arquivo total.pdf: 1366177 bytes, checksum: 1b02d1a5ce5861390070022558e311b0 (MD5) / Made available in DSpace on 2016-03-31T12:43:01Z (GMT). No. of bitstreams: 1 arquivo total.pdf: 1366177 bytes, checksum: 1b02d1a5ce5861390070022558e311b0 (MD5) Previous issue date: 2015-07-24 / This work is about the Rees algebra of a nite colength almost complete intersection ideal generated by forms of the same degree in a polynomial ring over a eld. We deal with two situations which are quite apart from each other: in the rst the forms are monomials in an unrestricted number of variables, while the second is for general binary forms. The essential goal in both cases is to obtain the depth of the Rees algebra. It is known that for such ideals the latter is rarely Cohen{Macaulay (i.e., of maximal depth). Thus, the question remains as to how far one is from the Cohen{Macaulay case. In the case of monomials one proves under certain restriction a conjecture of Vasconcelos to the e ect that the Rees algebra is almost Cohen{ Macaulay. At the other end of the spectrum, one proposes a proof of a conjecture of Simis on general binary forms, based on work of Huckaba{Marley and on a theorem concerning the Ratli {Rush ltration. Still within this frame, one states a couple of stronger conjectures that imply Simis conjecture, along with some solid evidence. / Este trabalho versa sobre a algebra de Rees de um ideal quase intersec cão completa, de cocomprimento nito, gerado por formas de mesmo grau em um anel de polinômios sobre um corpo. Considera-se duas situa c~oes inteiramente diversas: na primeira, as formas s~ao mon^omios em um n umero qualquer de vari aveis, enquanto na segunda, s~ao formas bin arias gerais. O objetivo essencial em ambos os casos e obter a profundidade da algebra de Rees. E conhecido que tal algebra e raramente Cohen{Macaulay (isto e, de profundidade m axima). Assim, a quest~ao que permanece e qua o distante são do caso Cohen{Macaulay. No caso de monômios prova-se, mediante certa restri cão, uma conjectura de Vasconcelos no sentido de que a algébra de Rees e quase Cohen {Macaulay. No outro caso extremo, estabelece-se uma prova de uma conjectura de Simis sobre formas bin arias gerais, baseada no trabalho de Huckaba{Marley e em um teorema sobre a ltera cão de Ratli {Rush. Al em disso, apresenta-se um par de conjecturas mais fortes que implicam a conjectura de Simis, juntamente com uma evidência s olida.

Les applications rationnelles sont des objets fondamentaux en géométrie algébrique. Elles sont utilisées pour décrire certains objets géométriques, tels que la représentation paramétrique d'une variété algébrique rationnelle. Plus récemment, les applications rationnelles sont apparues dans des contextes d'informatique pour l'ingénierie, dans le domaine de la modélisation de formes, en utilisant des méthodes de conception assistée par ordinateur pour les courbes et les surfaces. Des paramétrisations des courbes et des surfaces sont utilisées de manière intensive afin décrire des objets dans la modélisation géométrique, tel que structures des voitures, des avions. Par conséquent, l'étude des applications rationnelles est d'intérêt théorique dans la géométrie algébrique et l'algèbre commutative, et d'une importance pratique dans la modélisation géométrique. Ma thèse étudie les images et les fibres des applications rationnelles en relation avec les équations des algèbres de Rees et des algèbres symétriques. Dans la modélisation géométrique, il est important d'avoir une connaissance détaillée des propriétés géométriques de l'objet et de la représentation paramétrique avec lesquels on travaille. La question de savoir combien de fois le même point est peint (c'est-à-dire, correspond à des valeurs distinctes du paramètre), ne concerne pas seulement la variété elle-même, mais également la paramétrisation. Il est utile pour les applications de déterminer les singularités des paramétrisations. Dans les chapitres 2 et 3, on étudie des fibres d'une application rationnelle de P^m dans P^n qui est génériquement finie sur son image. Une telle application est définie par un ensemble ordonné de (n+1) polynômes homogènes de même degré d. Plus précisément, dans le chapitre 2, nous traiterons le cas des paramétrisations de surfaces rationnelles de P^2 dans P^3, et y donnons une borne quadratique en d pour le nombre de fibres de dimension 1 de la projection canonique de son graphe sur son image. Nous déduisons ce résultat d'une étude de la différence du degré initial entre les puissances ordinaires et les puissances saturées. Dans le chapitre 3, on affine et généralise les résultats sur les fibres du chapitre précédent. Plus généralement, nous établissons une borne linéaire en d pour le nombre de fibres (m-1)-dimensionnelles de la projection canonique de son graphe sur son image, en utilisant des idéaux de mineurs de la matrice jacobienne.Dans le chapitre 4, nous considérons des applications rationnelles dont la source est le produit de deux espaces projectifs.Notre principal objectif est d'étudier les critères de birationalité pour ces applications. Tout d'abord, un critère général est donné en termes du rang d'une couple de matrices connues sous le nom "matrices jacobiennes duales". Ensuite, nous nous concentrons sur des applications rationnelles de P^1 x P^1 vers P^2 en bidegré bas et fournissons de nouveaux critères de birationalité en analysant les syzygies des équations de définition de l'application; en particulier en examinant la dimension de certaines parties bigraduées du module de syzygies. Enfin, les applications de nos résultats au contexte de la modélisation géométrique sont discutées à la fin du chapitre. / Rational maps are fundamental objects in algebraic geometry. They are used to describe some geometric objects,such as parametric representation of rational algebraic varieties. Lately, rational maps appeared in computer-engineering contexts, mostly applied to shape modeling using computer-aided design methods for curves and surfaces. Parameterized algebraic curves and surfaces are used intensively to describe objects in geometric modeling, such as car bodies, airplanes.Therefore, the study of rational maps is of theoretical interest in algebraic geometry and commutative algebra, and of practical importance in geometric modeling. My thesis studies images and fibers of rational maps in relation with the equations of the symmetric and Rees algebras. In geometric modeling, it is of vital importance to have a detailed knowledge of the geometry of the object and of the parametric representation with which one is working. The question of how many times is the same point being painted (i.e., corresponds to distinct values of parameter), depends not only on the variety itself, but also on the parameterization. It is of interest for applications to determine the singularities of the parameterizations. In the chapters 2 and 3, we study the fibers of a rational map from P^m to P^nthat is generically finite onto its image. More precisely, in the second chapter, we will treat the case of parameterizations of algebraic rational surfaces. In this case, we give a quadratic bound in the degree of the defining equations for the number of one-dimensional fibers of the canonical projection of the graph of $\phi$ onto its image,by studying of the difference between the initial degree of ordinary and saturated powers of the base ideal. In the third chapter, we refine and generalize the results on fibers of the previous chapter.More generally, we establish a linear bound in the degree of the defining equations for the number of (m-1)-dimensional fibers of the canonical projection of its graph onto its image, by using ideals of minors of the Jacobian matrix.In the fourth chapter, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that came to be known as "Jacobian dual matrices". Then, we focus on rational maps from P^1 x P^1 to P^2 in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map, in particular by looking at the dimension of certain bigraded parts of the syzygy module. Finally, applications of our results to the context of geometric modeling are discussed at the end of the chapter.

Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes. / QC 20100622

One of the fundamental problems in commutative algebra and algebraic geometry is to understand the nature of the solution space of a system of multivariate polynomial equations over a field k, such as real or complex numbers. An important algorithmic tool in this study is the notion of Groebner bases (Buchberger (1965)). Given a system of polynomial equations, f1= 0,..., fm = 0, Groebner basis is a “canonical" generating set of the ideal generated by f1,...., fm, that can answer, constructively, many questions in computational ideal theory. It generalizes several concepts of univariate polynomials like resultants to the multivariate case, and answers decisively the ideal membership problem. The dimension of the solution set of an ideal I called the affine variety, an important concept in algebraic geometry, is equal to the Krull dimension of the corresponding coordinate ring, k[x1,...,xn]/I. Groebner bases were first introduced to compute k-vector space bases of k[x1,....,xn]/I and use that to characterize zero-dimensional solution sets. Since then, Groebner basis techniques have provided a generic algorithmic framework for computations in control theory, cryptography, formal verification, robotics, etc, that involve multivariate polynomials over fields. The main aim of this thesis is to study problems related to computational ideal theory over Noetherian commutative rings (e.g: the ring of integers, Z, the polynomial ring over a field, k[y1,…., ym], etc) using the theory of Groebner bases. These problems surface in many domains including lattice based cryptography, control systems, system-on-chip design, etc. Although, formal and standard techniques are available for polynomial rings over fields, the presence of zero divisors and non units make developing similar techniques for polynomial rings over rings challenging. Given a polynomial ring over a Noetherian commutative ring, A and an ideal I in A[x1,..., xn], the first fundamental problem that we study is whether the residue class polynomial ring, A[x1,..., xn]/I is a free A-module or not. Note that when A=k, the answer is always ‘yes’ and the k-vector space basis of k[x1,..., xn]/I plays an important role in computational ideal theory over fields. In our work, we give a Groebner basis characterization for A[x1,...,xn]/I to have a free A-module representation w.r.t. a monomial ordering. For such A-algebras, we give an algorithm to compute its A-module basis. This extends the Macaulay-Buchberger basis theorem to polynomial rings over Noetherian commutative rings. These results help us develop a theory of border bases in A[x1,...,xn] when the residue class polynomial ring is finitely generated. The theory of border bases is handled as two separate cases: (i) A[x1,...,xn]/I is free and (ii) A[x1,...,xn]/I has torsion submodules. For the special case of A = Z, we show how short reduced Groebner bases and the characterization for a free A-module representation help identify the cases when Z[x1,...,xn]/I is isomorphic to ZN for some positive integer N. Ideals in such Z-algebras are called ideal lattices. These structures are interesting since this means we can use the algebraic structure, Z[x1,...,xn]/I as a representation for point lattices and extend all the computationally hard problems in point lattice theory to Z[x1,...,xn]/I . Univariate ideal lattices are widely used in lattice based cryptography for they are a more compact representation for lattices than matrices. In this thesis, we give a characterization for multivariate ideal lattices and construct collision resistant hash functions based on them using Groebner basis techniques. For the construction of hash functions, we define a worst case problem, shortest substitution problem w.r.t. an ideal in Z[x1,...,xn], and establish hardness results for this problem. Finally, we develop an approach to compute the Krull dimension of A[x1,...,xn]/I using Groebner bases, when A is a Noetherian integral domain. When A is a field, the Krull dimension of A[x1,...,xn]/I has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. We introduce the notion of combinatorial dimension of A[x1,...,xn]/I and give a Groebner basis method to compute it for residue class polynomial rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A[x1,...,xn]/I. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Groebner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.