Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page

Serre’s Ex-Conjeture: Quillen-Suslin Theorem

RESUMEN: Este trabajo se dedica a explorar y presentar una de las famosas conjeturas de J. P. Serre, planteadas en la década de 1950, y la respuesta afirmativa que D. Quillen y A. A. Suslin dieron a dicha conjetura en 1976. Este resultado afirma que para módulos finitamente generados sobre anillos de polinomios con coeficientes en un dominio de ideales principales, las condiciones de módulo proyectivo y libre son equivalentes. Antes de mostrar la prueba de la ex-conjetura de Serre, vamos a introducir el concepto de propiedades locales, cuyo estudio contextualiza esta conjetura, así como algunos resultados instrumentales que son clave en la demostración. Uno de estos es el llamado Teorema de Quillen-Suslin, el cual constituye la pieza final que llevó a Quillen y a Suslin a resolver la conjetura de Serre. El principal objetivo de este trabajo es dar la prueba de Quillen del Teorema de Quillen-Suslin. Para concluir, presentaremos una aplicación de la ex-conjetura de Serre, con el propósito de mostrar que este abstracto resultado ha tenido una cierta trascendencia en el desarrollo posterior de las Matemáticas, especialmente de la Geometría Algebraica.ABSTRACT: This work is devoted to explore and present one of J. P. Serre’s famous conjectures, proposed in the decade of 1950, and the affirmative answer that D. Quillen and A. A. Suslin gave to this conjecture in 1976. This result states that for finitely generated modules over polynomial rings with coefficients in a principal ideal domain, the conditions of projective and free module are equivalent. Before showing the proof of Serre’s ex-conjecture, we will introduce the concept of local properties, whose study constitutes the context of this conjecture, as well as some instrumental theorems that are key in the proof. One of these is the so called Quillen-Suslin Theorem, and constitutes the final piece that led Quillen and Suslin to the solution of Serre’s conjecture. The main goal of this work is to give Quillen’s proof of Quillen-Suslin Theorem. Finally, we will present an application of Serre’s ex-conjeture, in order to show that this abstract result has had a certain trascendence in the posterior development of Mathematics, particularly in Algebraic Geometry.Grado en Matemática

On Flows, Paths, Roots, and Zeros

This thesis has two parts; in the first of which we give new results for various network flow problems. (1) We present a novel dual ascent algorithm for min-cost flow and show that an implementation of it is very efficient on certain instance classes. (2) We approach the problem of numerical stability of interior point network flow algorithms by giving a path following method that works with integer arithmetic solely and is thus guaranteed to be free of any nu-merical instabilities. (3) We present a gradient descent approach for the undirected transship-ment problem and its special case, the single source shortest path problem (SSSP). For distrib-uted computation models this yields the first SSSP-algorithm with near-optimal number of communication rounds. The second part deals with fundamental topics from algebraic computation. (1) We give an algorithm for computing the complex roots of a complex polynomial. While achieving a com-parable bit complexity as previous best results, our algorithm is simple and promising to be of practical impact. It uses a test for counting the roots of a polynomial in a region that is based on Pellet's theorem. (2) We extend this test to polynomial systems, i.e., we develop an algorithm that can certify the existence of a k-fold zero of a zero-dimensional polynomial system within a given region. For bivariate systems, we show experimentally that this approach yields signifi-cant improvements when used as inclusion predicate in an elimination method.Im ersten Teil dieser Dissertation präsentieren wir neue Resultate für verschiedene Netzwerkflussprobleme. (1)Wir geben eine neue Duale-Aufstiegsmethode für das Min-Cost-Flow- Problem an und zeigen, dass eine Implementierung dieser Methode sehr effizient auf gewissen Instanzklassen ist. (2)Wir behandeln numerische Stabilität von Innere-Punkte-Methoden fürNetwerkflüsse, indem wir eine solche Methode angeben die mit ganzzahliger Arithmetik arbeitet und daher garantiert frei von numerischen Instabilitäten ist. (3) Wir präsentieren ein Gradienten-Abstiegsverfahren für das ungerichtete Transshipment-Problem, und seinen Spezialfall, das Single-Source-Shortest-Problem (SSSP), die für SSSP in verteilten Rechenmodellen die erste mit nahe-optimaler Anzahl von Kommunikationsrunden ist. Der zweite Teil handelt von fundamentalen Problemen der Computeralgebra. (1) Wir geben einen Algorithmus zum Berechnen der komplexen Nullstellen eines komplexen Polynoms an, der eine vergleichbare Bitkomplexität zu vorherigen besten Resultaten hat, aber vergleichsweise einfach und daher vielversprechend für die Praxis ist. (2)Wir erweitern den darin verwendeten Pellet-Test zum Zählen der Nullstellen eines Polynoms auf Polynomsysteme, sodass wir die Existenz einer k-fachen Nullstelle eines Systems in einer gegebenen Region zertifizieren können. Für bivariate Systeme zeigen wir experimentell, dass eine Integration dieses Ansatzes in eine Eliminationsmethode zu einer signifikanten Verbesserung führt