Let K be a field and Q a finite directed multi-graph. In this paper I classify all path algebras KQ and admissible orders with the property that all of their finitely generated ideals have finite Groebner bases. / Master of Science

The first half of this thesis is devoted to the study of finite polynomial maps en --4 en and the use of Grobner bases to determine if a given map is finite. We begin by examining those maps which have quasihomogeneous components, and give a simple condition for such maps to be finite. This condition is extended to those maps which are quasihomogeneous as above, but with extra lower order terms. Next, we give a general criterion for testing the finiteness of a given polynomial map and an implementation in the Maple computer algebra system. Our next step is to generalize our results to regular maps between affine varieties. Again, a finiteness criterion is given, plus its implementation in Maple. Lastly in this half, we consider the trace bilinear form associated with a finite map and show how it may be used to find real roots of a polynomial system. The second half of the thesis is concerned with the study of G-variant map germs, which commute with the action of a finite group G on the source and target spaces. We give a relation between the G-variant degree associated with a map germ, bilinear forms on the local algebra and preimages of zero under a perturbation of the original map. We look at both the complex and real affine space situation. We then give the equivalent results when we do not have a 'good' deformation of the map, when we have two groups acting and when we use modular representations. Next, we give an invariant of G-variant maps which is stronger than G-degree, based upon a lattice of vector subspaces. Finally, we examine the structure of the class of G-variant maps and consider criteria for maps to have 'good' deformations and to be finite. We then give ways of determining generators for the class of maps by generalizing theorems of Noether and Molien.

Nesta dissertação estudamos algumas aplicações da teoria das bases de Gröbner, visando principalmente a utilização dessas técnicas na teoria de códigos. Apresentamos um algoritmo para obter a base de Gröbner reduzida do ideal de um conjunto finito de pontos, e descrevemos um método para encontrar aproximações de Padé de polinômios multivariados. Terminamos apresentando o procedimento desenvolvido por J. Farr e S. Gao para a construção e decodificação de códigos lineares via bases de Gröbner. / In this master thesis we study some applications of Grobner bases theory, aiming using these techniques in coding theory. We present an algorithm for computing the reduced Grobner basis of the vanishing ideal of a finite set of points, and describe a method for finding Padé approximations of multivariate polynomials. We finish presenting the procedure developed by J. Farr and S. Gao for construction and decoding of linear codes via Gröbner bases.

Nesta dissertação estudamos algumas aplicações da teoria das bases de Gröbner, visando principalmente a utilização dessas técnicas na teoria de códigos. Apresentamos um algoritmo para obter a base de Gröbner reduzida do ideal de um conjunto finito de pontos, e descrevemos um método para encontrar aproximações de Padé de polinômios multivariados. Terminamos apresentando o procedimento desenvolvido por J. Farr e S. Gao para a construção e decodificação de códigos lineares via bases de Gröbner. / In this master thesis we study some applications of Grobner bases theory, aiming using these techniques in coding theory. We present an algorithm for computing the reduced Grobner basis of the vanishing ideal of a finite set of points, and describe a method for finding Padé approximations of multivariate polynomials. We finish presenting the procedure developed by J. Farr and S. Gao for construction and decoding of linear codes via Gröbner bases.

Nesta dissertação estudamos algumas aplicações da teoria das bases de Gröbner, visando principalmente a utilização dessas técnicas na teoria de códigos. Apresentamos um algoritmo para obter a base de Gröbner reduzida do ideal de um conjunto finito de pontos, e descrevemos um método para encontrar aproximações de Padé de polinômios multivariados. Terminamos apresentando o procedimento desenvolvido por J. Farr e S. Gao para a construção e decodificação de códigos lineares via bases de Gröbner. / In this master thesis we study some applications of Grobner bases theory, aiming using these techniques in coding theory. We present an algorithm for computing the reduced Grobner basis of the vanishing ideal of a finite set of points, and describe a method for finding Padé approximations of multivariate polynomials. We finish presenting the procedure developed by J. Farr and S. Gao for construction and decoding of linear codes via Gröbner bases.

Ce mémoire se veut une étude détaillée de ce que sont les bases de Groebner, de la manière dont on les calcule et dans quels cas elles sont utiles et utilisées. Un éventail de définitions, de théorèmes, de lemmes et de propositions sont énoncés et démontrés afin que les lecteurs, lectrices, intéressé(e)s, puissent avoir les ressources nécessaires leur permettant de comprendre vraiment ce que sont les bases de Groebner. Ce travail propose également une définition précise de ce que sont les ordres monomiaux et élabore une formulation claire de leur classification. Ce mémoire donne, en plus, une description des algorithmes sous forme de procédures, programmés en utilisant le logiciel Maple 10 qui sont mis en annexe. Tous les algorithmes, décrits en pseudo-code, ont été programmés de manière naïve, c'est-à-dire, sans astuce de programmation afin d'en réduire le temps d'exécution ou l'espace mémoire occupé. Cela afin de faire voir aux lecteurs, lectrices, intéressé(e)s, comment se font les calculs. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Anneau, Idéal, Base de Groebner, Module, Ordre monomial, Ordre monoïdal.

Commutative algebra finds its roots in invariant theory and the connection is drawn from a modern standpoint. The Hilbert Basis Theorem and the Nullstellenstatz were considered lemmas for classical invariant theory. The Groebner basis is a modern tool used and is implemented with the computer algebra system Mathematica. Number 14 of Hilbert\'s 23 problems is discussed along with the notion of invariance under a group action of GLn(C). Computational difficulties are also discussed in reference to Groebner bases and Invariant theory.The straitening law is presented from a Groebner basis point of view and is motivated as being a key piece of machinery in proving First Fundamental Theorem of Invariant Theory. / Master of Science

La principal contribución de esta tesis es la vinculación entre el Álgebra y la Teoría de Bifurcaciones. Se estudian sistemas dinámicos que puedan expresarse con representaciones po-linomiales, y se utilizan herramientas algebraicas para realizar análisis, síntesis y control sobre estos modelos matemáticos. El método de las formas normales es una técnica clásica de estudio en teoría de bifurcaciones. El objetivo es capturar los elementos fundamentales de la solución de un sistema, obte-niendo como resultado una estructura polinomial en las ecua-ciones. Por medio del polinomio de la forma normal se pueden determinar amplitud, frecuencia, estabilidad y multiplicidad de las oscilaciones, también conocidas como ciclos límites. El método también permite realizar control de bifurcaciones, es decir diseñar un controlador que pueda modifcar las caracte-rísticas de bifurcación de un dado sistema, alcanzando ciertos comportamientos dinámicos más deseables. Algunos de los objetivos podrían ser: a) cambiar el valor del parámetro de un punto de bifurcación, b) estabilizar una solución o una rama de soluciones bifurcadas, c) modificar la multiplicidad de esta-do estacionario, de soluciones periódicas o de atractores, d) modificar la frecuencia y amplitud de las soluciones emergen-tes de una bifurcación, etc. La descripción clásica de siste-mas dinámicos en el espacio de estados se enfoca desde un punto de vista algebraico. Las dinámicas se pueden represen-tar como relaciones polinómicas entre las variables de estado y sus primeras derivadas. Entonces, esta descripción alge-braica constituye el conjunto de todos los polinomios que se anulan, cuando las variables toman los valores de cualquier trayectoria del sistema en estudio. Esta representación está directamente conectada con los conceptos algebraicos de ideales y variedades, y permite trabajar con algoritmos y herramientas algebraicas como por ejemplo las bases de Gro-ebner. El método de síntesis no sólo muestra que es posible diseñar osciladores y controladores, sino también obtener formas normales, y un conjunto de polinomios que genera una familia de sistemas dinámicos, todos ellos con una órbita en común entre sus posibles soluciones. Estos problemas se re-suelven en forma analítica con métodos simbólicos y también se usan técnicas numéricas. A pesar de que los métodos simbólicos proveen una solución más general, aún se encuen-tran limitados a resolver problemas de tamaño modesto. Una alternativa es usar ambos métodos en forma complementaria. Existen programas específicos tanto simbólicos como numé-ricos, y se utilizan para la eliminación de variables en sistemas de ecuaciones polinómicas y luego en forma numérica, cálculo de raíces, simulaciones y continuación de bifurcaciones. El impacto y alcance del uso de las bases de Groebner en sistemas dinámicos resalta sobre los conocimientos ya de-sarrollados como un paso hacia la ampliación de futuras investigaciones. / The main contribution of this thesis is the exploration of relationships between Algebra and Bifurcation Theory. Pro-blems such as bifurcation control and synthesis of dynamical systems based on polynomial representations are studied, and algebraic tools are used for analysis, synthesis and control of these mathematical models. Using the normal form method, a classic technique for studying bifurcation theory, dynamical behaviors can be captured by a set of polynomial equations, and the stability, amplitude, frequency and multiplicity of the periodic solutions are analyzed using polynomials. This approach is also suitable for bifurcation control, i.e. the design of a controller to modify the bifurca-tion characteristics of a given system to obtain a more desira-ble dynamic behavior. In this way, it is possible to delay the appearance of bifurcations, to stabilize a solution or a branch of bifurcation solutions, to alter its multiplicity, to modify the frequency and amplitude of the solutions emerging from a bi-furcation, etc. The classical state-space description of a dynamical system can also be treated within this algebraic framework. The system is represented by a set of polynomial equations involving the variables and their derivatives that vanishes over any path of the system under study. This repre-sentation is linked with the algebraic concepts of ideal and varieties, and can be handled with tools such as Groebner bases. Controller synthesis, oscillator design and derivation of normal forms can also be achieved with Groebner basis. As a result of the design procedure, a set of polynomials that generates a family of dynamical systems sharing a common orbit among their possible trajectories is obtained. These pro-blems can be solved using symbolic or numerical methods, but even for systems of modest size, the complexity of the symbolic solutions is overwhelming. An alternative explored in this thesis is to employ numerical and symbolic methods complementarily, for example eliminating variables in systems of polynomial equations symbolically, and then computing the roots, continuation of bifurcations and simulations using nume-rical techniques. The impact and scope of Groebner bases in dynamical systems are highlighted with respect to what has already been acomplished as a stepping stone for future research.

We compute the Groebner basis of a system of polynomial equations related to the Jacobian conjecture using a recursive formula for the Catalan numbers. / En este artículo calculamos la base de Groebner de un sistema polinomial de ecuaciones relacionada con la conjetura del jacobiano utilizando una fórmula recursiva para los numeros de Catalan.

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal without changing the variety is a powerful tool in determining a variety. In general, the quotient and remainder on division of polynomials in more than one variable are not unique. One property of a Groebner basis is that it yields a unique remainder on division. To prove geometric theorems algebraically, we first express the hypotheses and conclusions as polynomials. Then, with the aid of a computer, apply the Groebner Basis Algorithm to determine if the conclusion polynomial(s) vanish on the same variety as the hypotheses.