Rational Algebraic Curves A Computer Algebra Approach
Algebraic curves and surfaces are an old topic of geometric and algebraic investigation. They have found applications for instance in ancient and m- ern architectural designs, in number theoretic problems, in models of b- logical shapes, in error-correcti
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J. Rafael Sendra Franz Winkler Sonia Pérez-Díaz
Rational Algebraic Curves A Computer Algebra Approach
Algorithms and Computation in Mathematics • Volume 22 Editors Arjeh M. Cohen Henri Cohen David Eisenbud Michael F. Singer Bernd Sturmfels
J. Rafael Sendra Franz Winkler Sonia Pérez-Díaz
Rational Algebraic Curves A Computer Algebra Approach
With 24 Figures and 2 Tables
Authors J. Rafael Sendra Departamento de Matemáticas Universidad de Alcalá 28871 Alcalá de Henares, Madrid Spain E-mail: [email protected]
Franz Winkler RISC-Linz J. Kepler Universität Linz 4040 Linz Austria E-mail: [email protected]
Sonia Pérez-Díaz Departamento de Matemáticas Universidad de Alcalá 28871 Alcalá de Henares, Madrid Spain E-mail: [email protected]
Library of Congress Control Number: 2007932190 Mathematics Subject Classification (2000): 14H50, 14M20, 14Q05, 68W30
ISSN 1431-1550 ISBN 978-3-540-73724-7 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speci cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro lm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speci c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author and SPi using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
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Preface
Algebraic curves and surfaces are an old topic of geometric and algebraic investigation. They have found applications for instance in ancient and modern architectural designs, in number theoretic problems, in models of biological shapes, in error-correcting codes, and in cryptographic algorithms. Recently they have gained additional practical importance as central objects in computer-aided geometric design. Modern airplanes, cars, and household appliances would be unthinkable without the computational manipulation of algebraic curves and surfaces. Algebraic curves and surfaces combine fascinating mathematical beauty with challenging computational complexity and wide spread practical applicability. In this book we treat only algebraic curves, although many of the results and methods can be and in fact have been generalized to surfaces. Being the solution loci of algebraic, i.e., polynomial, equations in two variables, plane algebraic curves are well suited for b
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