Roots of unity as quotients of two conjugate algebraic numbers

Let α be an algebraic number of degree d ≥ 2 over Q. Suppose for some pairwise coprime positive integers n1,… ,nr we have deg(αnj) < d for j=1,…,r, where deg(αn)=d for each positive proper divisor n of nj. We prove that then φ(n1 … nr) ≤ d, where φ stands for the Euler totient function. In particular, if nj=pj, j=1,…,r, are any r distinct primes satisfying deg(αpj) < d, then the inequality (p1-1)… (pr-1) ≤ d holds, and therefore r ≪ log d/log log d for d ≥ 3. This bound on r improves that of Dobrowolski r ≤ log d/log 2 proved in 1979 and is best possible

Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

On methods of computing galois groups and their implementations in MAPLE.

by Tang Ko Cheung, Simon.Thesis date on t.p. originally printed as 1997, of which 7 has been overwritten as 8 to become 1998.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 95-97).Chapter 1 --- Introduction --- p.5Chapter 1.1 --- Motivation --- p.5Chapter 1.1.1 --- Calculation of the Galois group --- p.5Chapter 1.1.2 --- Factorization of polynomials in a finite number of steps IS feasible --- p.6Chapter 1.2 --- Table & Diagram of Transitive Groups up to Degree 7 --- p.8Chapter 1.3 --- Background and Notation --- p.13Chapter 1.4 --- Content and Contribution of THIS thesis --- p.17Chapter 2 --- Stauduhar's Method --- p.20Chapter 2.1 --- Overview & Restrictions --- p.20Chapter 2.2 --- Representation of the Galois Group --- p.21Chapter 2.3 --- Groups and Functions --- p.22Chapter 2.4 --- Relative Resolvents --- p.24Chapter 2.4.1 --- Computing Resolvents Numerically --- p.24Chapter 2.4.2 --- Integer Roots of Resolvent Polynomials --- p.25Chapter 2.5 --- The Determination of Galois Groups --- p.26Chapter 2.5.1 --- Searching Procedures --- p.26Chapter 2.5.2 --- "Data: T(x1,x2 ,... ,xn), Coset Rcpresentatives & Searching Diagram" --- p.27Chapter 2.5.3 --- Examples --- p.32Chapter 2.6 --- Quadratic Factors of Resolvents --- p.35Chapter 2.7 --- Comment --- p.35Chapter 3 --- Factoring Polynomials Quickly --- p.37Chapter 3.1 --- History --- p.37Chapter 3.1.1 --- From Feasibility to Fast Algorithms --- p.37Chapter 3.1.2 --- Implementations on Computer Algebra Systems --- p.42Chapter 3.2 --- Squarefree factorization --- p.44Chapter 3.3 --- Factorization over finite fields --- p.47Chapter 3.4 --- Factorization over the integers --- p.50Chapter 3.5 --- Factorization over algebraic extension fields --- p.55Chapter 3.5.1 --- Reduction of the problem to the ground field --- p.55Chapter 3.5.2 --- Computation of primitive elements for multiple field extensions --- p.58Chapter 4 --- Soicher-McKay's Method --- p.60Chapter 4.1 --- "Overview, Restrictions and Background" --- p.60Chapter 4.2 --- Determining cycle types in GalQ(f) --- p.62Chapter 4.3 --- Absolute Resolvents --- p.64Chapter 4.3.1 --- Construction of resolvent --- p.64Chapter 4.3.2 --- Complete Factorization of Resolvent --- p.65Chapter 4.4 --- Linear Resolvent Polynomials --- p.67Chapter 4.4.1 --- r-sets and r-sequences --- p.67Chapter 4.4.2 --- Data: Orbit-length Partitions --- p.68Chapter 4.4.3 --- Constructing Linear Resolvents Symbolically --- p.70Chapter 4.4.4 --- Examples --- p.72Chapter 4.5 --- Further techniques --- p.72Chapter 4.5.1 --- Quadratic Resolvents --- p.73Chapter 4.5.2 --- Factorization over Q(diac(f)) --- p.73Chapter 4.6 --- Application to the Inverse Galois Problem --- p.74Chapter 4.7 --- Comment --- p.77Chapter A --- Demonstration of the MAPLE program --- p.78Chapter B --- Avenues for Further Exploration --- p.84Chapter B.1 --- Computational Galois Theory --- p.84Chapter B.2 --- Notes on SAC´ؤSymbolic and Algebraic Computation --- p.88Bibliography --- p.9

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

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