In this paper, we study primary decomposition of any proper submodule N of a module M over a noetherian ring R. We start by briefly discussing basic facts about the very well known case where M is a finitely generated module over a Noetherian ring R, then we proceed to discuss the general case where M is any module over a Noetherian ring R. We put a lot of focus on the associated primes that occur with the primary decomposition, essentially studying their uniqueness and their relation to the associated primes of M/N.

This paper presents the theory of Secondary Representation of modules over a commutative ring and their Attached Primes; introduced in 1973 by I. MacDonald as a dual to the important theory of associated primes and primary decomposition in commutative algebra. The paper explores many of the basic aspects of the theory of primary decomposition and associated primes of modules in the hopes to delineate and motivate the construction of a secondary representation, when possible. The thesis discusses the results of the uniqueness of representable modules and their attached primes, and, in particular, the existence of a secondary representation for Artinian modules. It concludes with some interesting examples of both secondary and representable modules, highlighting the consequences of the results thus established.

We describe the space of polynomial fields tangent to a given an algebraic curve. / Se da una descripción del espacio de campos polinomiales tangentes a una curva algebraica dada.

Biological systems are commonly represented using networks consisting of interactions between various elements in the system. Reverse engineering, a method of mathematical modeling, is used to recover how the elements in the biological network are connected. These connections are encoded using wiring diagrams, which are directed graphs that describe how elements in a network affect one another. A signed wiring diagram provides additional information about the interactions between elements relating to activation and inhibition. Due to cost concerns, it is optimal to gain insight into biological networks with as few experiments and data as possible. Minimal wiring diagrams identify the minimal sets of variables for which a model that fits the data exists. Previously established algorithms to compute possible minimal wiring diagrams rely on the primary decomposition of ideals in polynomial rings. Stanley-Reisner theory provides a one-to-one correspondence between squarefree monomial ideals and abstract simplicial complexes. In this work, we use this correspondence to determine conditions under which a given set of inputs is guaranteed to have a unique signed minimal wiring diagram, regardless of the output assignment.

No description available.

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.

First in this dissertation we make a brief overview about basic tools of commutative algebra required for understanding the rest of the text. Then, we present the definition of symbolic powers and we discuss their basic properties, mainly emphasizing questions such as primary decomposition and calculation of generators. We conclude this work by showing actual results that relate the symbolic powers with other notions in commutative algebra and algebraic geometry. / Nesta dissertação fazemos inicialmente um breve apanhado sobre ferramentas básicas de álgebra comutativa úteis para o entendimento do resto do texto. Em seguida, apresentamos a definição de potências simbólicas e discutimos suas propriedades mais elementares, destacando sobretudo questões como decomposição primária e cálculo de geradores. Finalizamos o trabalho mostrando resultados atuais que relacionam as potências simbólicas com outras noções da álgebra comutativa e geometria algébrica.

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels'' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.