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Department of Mathematics CENTER FOR RESEARCH AND ADVANCED STUDIES OF THE NATIONAL POLYTHECHNIC INSTITUTE CAMPUS ZACATENCO DEPARTMENT OF MATHEMATICS Numerical Functions of Graded Ideals, Edge-Ideals of Digraphs and their Applications to Coding Theory Dissertation submitted by Carlos Eduardo Vivares Parra to obtain the Degree of DOCTOR OF SCIENCE IN THE SPECIALITY OF MATHEMATICS Thesis Advisor: Dr. Rafael Heraclio Villarreal Rodr´ıguez Mexico City February, 2019 CENTRO DE INVESTIGACION´ Y DE ESTUDIOS AVANZADOS DEL INSTITUTO POLITECNICO´ NACIONAL UNIDAD ZACATENCO DEPARTAMENTO DE MATEMATICAS´ Funciones Num´ericasde Ideales Graduados, Ideales de Aristas de Digr´aficasy sus Aplicaciones a la Teor´ıade C´odigos. Tesis que presenta Carlos Eduardo Vivares Parra para obtener el grado de DOCTOR EN CIENCIAS EN LA ESPECIALIDAD DE MATEMATICAS´ Asesor de Tesis: Dr. Rafael Heraclio Villarreal Rodr´ıguez Ciudad de Mexico´ Febrero, 2019 Contents Abstract 7 Resumen 9 Introduction 11 Acknowledgments 23 Chapter 1. Primary Decompositions and Graded Modules 25 1.1. Module theory 25 1.2. Graded modules and Hilbert polynomials 53 1.2.1. Graded primary decomposition 54 1.2.2. The Hilbert–Serre and Hilbert Theorems 56 1.3. Multiplicities of modules over local rings 60 Chapter 2. Hilbert Functions and Vanishing Ideals in Affine and Projective Varieties 63 2.1. Monomial ideals 63 2.2. Grobner¨ bases 68 2.3. Hilbert functions 73 2.4. The footprint of an ideal 88 2.5. Computing zeros of polynomials 89 Chapter 3. Reed–Muller-Type Codes 95 3.1. Projective Reed–Muller-type codes 95 3.2. Regularity and minimum distance 97 3.3. Affine Reed–Muller-type codes 99 Chapter 4. Generalized Minimum Distance Functions 103 4.1. Generalized Hamming weights and commutative algebra 103 4.2. Computing the number of points of a variety 106 4.3. Generalized minimum distance function of a graded ideal 109 4.4. An integer inequality 113 4.5. Second generalized Hamming weight 117 5 6 CONTENTS 4.6. Generalized Hamming weights of affine cartesian codes 121 Chapter 5. Cohen-Macaulay vertex-weighted digraphs 125 5.1. Irreducible decompositions and symbolic powers 125 5.2. Cohen–Macaulay weighted oriented trees 134 Chapter 6. Depth and regularity of monomial ideals via polarization and combinatorial optimization 145 6.1. Depth and regularity of monomial ideals via polarization 145 6.2. Depth and regularity locally at each variable 155 6.3. Edge ideals of clutters with non increasing depth 160 6.4. Edge ideals of graphs 163 Chapter 7. Conclusions and future work 171 Bibliography 173 Index of Definitions 181 Abstract In the first part we introduce the fundamental tools of commutative algebra and al- gebraic geometry needed for the study of the coding theory. Combining these tools with combinatorics we obtain formulas for the most important parameter of a code, the mini- mum distance. Next we study monomial ideals, Grobner¨ bases and the footprint of an ideal, projec- tive closures, vanishing ideals, and Hilbert functions. The role of Hilbert functions and vanishing ideals in affine and projective varieties is discussed here. The number of zeros that a homogeneous polynomial has in any given finite set of points in an affine or pro- jective space is expressed in terms of vanishing ideals and the notion of degree. Also we study the families of projective and affine Reed–Muller-type codes and their connection to vanishing ideals and Hilbert functions. In the chapter 4, we explore the r-th generalized minimum distance function (gmd function for short) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. If X is a set of projective points over a finite field and I(X) is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I(X) are equal to the r-th generalized Hamming weight of the corresponding Reed- Muller-type code CX(d). We show that the r-th generalized footprint function of I(X) is a lower bound for the r-th generalized Hamming weight of CX(d). As an application to coding theory we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine cartesian code. In the chapter 5, we give an effective characterization of the Cohen–Macaulay vertex- weighted oriented trees and forests. For transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen–Macaulay and satisfy Alexander duality. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of the normality of its primary components. Finally in Chapter 6, we use polarization to study the behavior of the depth and regu- larity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the 7 8 ABSTRACT highest powers of the variable xi occurring in the minimal generating set of I, and exam- ine the depth and regularity of powers of edge ideals of clutters using combinatorial op- timization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min- cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. As a consequence edge ideals of unmixed bipartite graphs have non-increasing depth. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter. Resumen En la primera parte introducimos las herramientas fundamentales del algebra´ conmu- tativa y geometr´ıa algebraica necesarias para el estudio de la teor´ıa de Codigos.´ Combi- nando estas herramientas con combinatoria obtenemos formulas´ para el parametro´ mas´ importante de un codigo,´ la m´ınima la distancia. A continuacion´ estudiamos los ideales monomiales, las bases de Grobner¨ y la huella de un ideal, la clausura proyectiva, ideales de anulacion,´ y funciones de Hilbert. El numero´ de ceros que un polinomio homogeneo´ tiene en cualquier conjunto finito de puntos en un espacio af´ın o proyectivo se expresa en terminos´ de ideales de anulacion´ y la nocion´ de grado. Tambien´ estudiamos las familias de codigos´ proyectivos y afines de tipo Reed-Muller y su conexion.´ con los ideales de anulacion´ y las funciones de Hilbert. En el cap´ıtulo 4, exploramos la r-esima´ funcion´ de m´ınima distancia generalizada (funcion´ gmd para abreviar) y la correspondiente funcion´ de huella generalizada de un ideal graduado en un anillo polinomial sobre un campo. Si X es un conjunto puntos proyectivos sobre un campo finito, I(X) su ideal de anulacion,´ demostramos que la funcion´ gmd y la funcion´ de Vasconcelos de I(X) son iguales al r- esimo´ peso de Hamming generalizado del correspondiente codigo´ de tipo Reed-Muller CX(d). Mostramos que la r-esima´ funcion´ huella generalizada de I(X) es un l´ımite infe- rior para el r-esimo´ peso de Hamming generalizado de CX(d). Como una aplicacion´ para la teor´ıa de codigos´ mostramos una formula´ expl´ıcita y una formula´ combinatoria para el segundo peso generalizado de Hamming de un codigo´ cartesiano af´ın. En el cap´ıtulo 5, damos una caracterizacion´ efectiva de los arboles´ y bosques orientados con peso Cohen- Macaulay. Para los graficos´ orientados con peso mostramos que la dualidad de Alexander se tiene. Finalmente en el Cap´ıtulo 6, usamos la polarizacion´ para estudiar el compor- tamiento de la profundidad y la regularidad de un ideal monomial I, localmente en una variable xi, cuando bajamos el grado de todas las potencias mas´ altas de la variable xi que esta´ en el conjunto m´ınimo generador de I, y examinamos la profundidad y la regulari- dad de las potencias ideales de aristas de hipergraficas´ utilizando tecnicas´ combinatoria. Si un ideal de aristas de una hipergrafica´ no mezclada con la propiedad min-cut max-flow, 9 10 RESUMEN demostramos que las potencias de I tienen profundidad no creciente y regularidad no de- creciente. Como consecuencia, los ideales de aristas de graficas´ bipartitas no mezcladas tienen profundidad no decreciente. Introduction This thesis studies certain numerical functions coming from graded ideals (e.g., gen- eralized minimum distance functions, regularity and depth) and certain algebraic proper- ties of graded ideals and their symbolic and ordinary powers (e.g., Complete intersection, Cohen-Macaulay, normality, unmixed, non-increasing depth, non-decreasing regularity). We also study the irreducible decomposition of a monomial ideal and the algebraic prop- erties of edge ideals of oriented graphs. This work begins the study of the generalized minimum distance function—a general numerical function of a graded ideals—of which the r-th generalized Hamming weight of a Reed-Muller type code is the simplest, but also the typical case. This is naturally connected to coding theory and to algebraic geometry over finite fields, i.e., to projective varieties and vanishing ideals over finite fields. The connection of the r-th generalized minimum distance function to coding theory comes from the observation that for vanishing ideals over finite fields and r = 1, this function is the minimum distance of the corresponding Reed-Muller type code. Coding theory is the study of error-correcting codes and their associated mathematics. An error- correcting code is used to encode information that will be transmitted through a noisy communication channel, in such a way that the original message can be recovered even though errors have occurred during the process. In a few words, the information to be sent turns into a binary string, then it is transmitted through a telephone, radio, satellite, etc., and when it reaches its destination the binary string may not have the same digits, because of human errors, electronic failures, weather, etc.
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