Algorithms and Computation in Mathematics • Volume 10 Editors Arjeh M Cohen Henri Cohen David Eisenbud Michael F Singer Bernd Sturmfels CuuDuongThanCong.com Saugata Basu Richard Pollack Marie-Franỗoise Roy Algorithms in Real Algebraic Geometry Second Edition With 37 Figures 123 CuuDuongThanCong.com Saugata Basu Georgia Institute of Technology School of Mathematics Atlanta, GA 30332-0160 USA e-mail: saugata@math.gatech.edu Richard Pollack Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 USA e-mail: pollack@cims.nyu.edu Marie-Franỗoise Roy IRMAR Campus de Beaulieu Universitộ de Rennes I 35042 Rennes cedex France e-mail: Marie-Francoise.Roy@univ-rennes1.fr Library of Congress Control Number: 2006927110 Mathematics Subject Classification (2000): 14P10, 68W30, 03C10, 68Q25, 52C45 ISSN 1431-1550 ISBN-10 3-540-33098-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33098-1 Springer Berlin Heidelberg New York ISBN 3-540-00973-6 1st edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 © Springer-Verlag Berlin Heidelberg 2003, 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typeset by the authors using a Springer LATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper CuuDuongThanCong.com 46/3100YL - Table of Contents Introduction Algebraically Closed Fields 1.1 Definitions and First Properties 1.2 Euclidean Division and Greatest Common Divisor 1.3 Projection Theorem for Constructible Sets 1.4 Quantifier Elimination and the Transfer Principle 1.5 Bibliographical Notes 11 11 14 20 25 27 Real Closed Fields 2.1 Ordered, Real and Real Closed Fields 2.2 Real Root Counting 2.2.1 Descartes’s Law of Signs and the Budan-Fourier Theorem 2.2.2 Sturm’s Theorem and the Cauchy Index 2.3 Projection Theorem for Algebraic Sets 2.4 Projection Theorem for Semi-Algebraic Sets 2.5 Applications 2.5.1 Quantifier Elimination and the Transfer Principle 2.5.2 Semi-Algebraic Functions 2.5.3 Extension of Semi-Algebraic Sets and Functions 2.6 Puiseux Series 2.7 Bibliographical Notes 29 29 44 44 52 57 63 69 69 71 72 74 81 Semi-Algebraic Sets 3.1 Topology 3.2 Semi-algebraically Connected Sets 3.3 Semi-algebraic Germs 83 83 86 87 CuuDuongThanCong.com VI Table of Contents 3.4 Closed and Bounded Semi-algebraic Sets 3.5 Implicit Function Theorem 3.6 Bibliographical Notes 93 94 99 Algebra 4.1 Discriminant and Subdiscriminant 4.2 Resultant and Subresultant Coefficients 4.2.1 Resultant 4.2.2 Subresultant Co efficients 4.2.3 Subresultant Co efficients and Cauchy Index 4.3 Quadratic Forms and Root Counting 4.3.1 Quadratic Forms 4.3.2 Hermite’s Quadratic Form 4.4 Polynomial Ideals 4.4.1 Hilbert’s Basis Theorem 4.4.2 Hilbert’s Nullstellensatz 4.5 Zero-dimensional Systems 4.6 Multivariate Hermite’s Quadratic Form 4.7 Projective Space and a Weak Bézout’s Theorem 4.8 Bibliographical Notes 101 101 105 105 110 113 119 119 127 132 132 136 143 149 153 157 Decomposition of Semi-Algebraic Sets 5.1 Cylindrical Decomposition 5.2 Semi-algebraically Connected Components 5.3 Dimension 5.4 Semi-algebraic Description of Cells 5.5 Stratification 5.6 Simplicial Complexes 5.7 Triangulation 5.8 Hardt’s Triviality Theorem and Consequences 5.9 Semi-algebraic Sard’s Theorem 5.10 Bibliographical Notes Elements of Topology 6.1 Simplicial Homology Theory 6.1.1 The Homology Groups of a Simplicial Complex 6.1.2 Simplicial Cohomology Theory 6.1.3 A Characterization of H1 in a Special Case 6.1.4 The Mayer-Vietoris Theorem 195 195 195 199 201 206 CuuDuongThanCong.com 159 159 168 170 172 174 181 183 186 191 194 Table of Contents VII 6.1.5 Chain Homotopy 6.1.6 The Simplicial Homology Groups Are Invariant Under Homeomorphism 6.2 Simplicial Homology of Closed and Bounded Semi-algebraic Sets 6.2.1 Definitions and First Properties 6.2.2 Homotopy 6.3 Homology of Certain Locally Closed Semi-Algebraic Sets 6.3.1 Homology of Closed Semi-algebraic Sets and of Sign Conditions 6.3.2 Homology of a Pair 6.3.3 Borel-Moore Homology 6.3.4 Euler-Poincaré Characteristic 6.4 Bibliographical Notes Quantitative Semi-algebraic Geometry 7.1 Morse Theory 7.2 Sum of the Betti Numbers of Real Algebraic Sets 7.3 Bounding the Betti Numbers of Realizations of Sign Conditions 7.4 Sum of the Betti Numbers of Closed Semi-algebraic Sets 7.5 Sum of the Betti Numbers of Semi-algebraic Sets 7.6 Bibliographical Notes Complexity of Basic Algorithms 8.1 Definition of Complexity 8.2 Linear Algebra 8.2.1 Size of Determinants 8.2.2 Evaluation of Determinants 8.2.3 Characteristic Polynomial 8.2.4 Signature of Quadratic Forms 8.3 Remainder Sequences and Subresultants 8.3.1 Remainder Sequences 8.3.2 Signed Subresultant Polynomials 8.3.3 Structure Theorem for Signed Subresultants 8.3.4 Size of Remainders and Subresultants 8.3.5 Specialization Properties of Subresultants 8.3.6 Subresultant Computation 8.4 Bibliographical Notes CuuDuongThanCong.com 209 213 221 221 223 226 226 228 231 234 236 237 237 256 262 268 273 280 281 281 292 292 294 299 300 301 301 303 307 314 316 317 322 VIII Table of Contents Cauchy Index and Applications 9.1 Cauchy Index 9.1.1 Computing the Cauchy Index 9.1.2 Bezoutian and Cauchy Index 9.1.3 Signed Subresultant Sequence and Cauchy Index on an Interval 9.2 Hankel Matrices 9.2.1 Hankel Matrices and Rational Functions 9.2.2 Signature of Hankel Quadratic Forms 9.3 Number of Complex Roots with Negative Real Part 9.4 Bibliographical Notes 10 Real Roots 10.1 Bounds on Roots 10.2 Isolating Real Roots 10.3 Sign Determination 10.4 Roots in a Real Closed Field 10.5 Bibliographical Notes 330 333 334 337 344 350 351 351 360 383 397 401 403 404 404 408 415 423 426 428 430 440 444 12 Polynomial System Solving 12.1 A Few Results on Gröbner Bases 12.2 Multiplication Tables 12.3 Special Multiplication Table 12.4 Univariate Representation 12.5 Limits of the Solutions of a Polynomial System 12.6 Finding Points in Connected Components of Algebraic Sets 12.7 Triangular Sign Determination 445 445 451 456 462 471 483 495 11 Cylindrical Decomposition Algorithm 11.1 Computing the Cylindrical Decomposition 11.1.1 Outline of the Method 11.1.2 Details of the Lifting Phase 11.2 Decision Problem 11.3 Quantifier Elimination 11.4 Lower Bound for Quantifier Elimination 11.5 Computation of Stratifying Families 11.6 Topology of Curves 11.7 Restricted Elimination 11.8 Bibliographical Notes CuuDuongThanCong.com 323 323 323 326 Table of Contents IX 12.8 Computing the Euler-Poincaré Characteristic of an Algebraic Set 498 12.9 Bibliographical Notes 503 13 Existential Theory of the Reals 13.1 Finding Realizable Sign Conditions 13.2 A Few Applications 13.3 Sample Points on an Algebraic Set 13.4 Computing the Euler-Poincaré Characteristic of Sign Conditions 13.5 Bibliographical Notes 505 506 516 519 14 Quantifier Elimination 14.1 Algorithm for the General Decision Problem 14.2 Quantifier Elimination 14.3 Local Quantifier Elimination 14.4 Global Optimization 14.5 Dimension of Semi-algebraic Sets 14.6 Bibliographical Notes 533 534 547 551 557 558 562 15 Computing Roadmaps and Connected Components of Algebraic Sets 15.1 Pseudo-critical Values and Connectedness 15.2 Roadmap of an Algebraic Set 15.3 Computing Connected Components of Algebraic Sets 15.4 Bibliographical Notes 528 532 563 564 568 580 592 16 Computing Roadmaps and Connected Components of Semialgebraic Sets 593 16.1 Special Values 593 16.2 Uniform Roadmaps 601 16.3 Computing Connected Components of Sign Conditions 608 16.4 Computing Connected Components of a Semi-algebraic Set 614 16.5 Roadmap Algorithm 617 16.6 Computing the First Betti Number of Semi-algebraic Sets 627 16.7 Bibliographical Notes 633 References 635 Index of Notation 645 Index 655 CuuDuongThanCong.com Introduction Since a real univariate polynomial does not always have real roots, a very natural algorithmic problem, is to design a method to count the number of real roots of a given polynomial (and thus decide whether it has any) The “real root counting problem” plays a key role in nearly all the “algorithms in real algebraic geometry” studied in this book Much of mathematics is algorithmic, since the proofs of many theorems provide a finite procedure to answer some question or to calculate something A classic example of this is the proof that any pair of real univariate polynomials (P , Q) have a greatest common divisor by giving a finite procedure for constructing the greatest common divisor of (P , Q), namely the euclidean remainder sequence However, different procedures to solve a given problem differ in how much calculation is required by each to solve that problem To understand what is meant by “how much calculation is required”, one needs a fuller understanding of what an algorithm is and what is meant by its “complexity” This will be discussed at the beginning of the second part of the book, in Chapter The first part of the book (Chapters through 7) consists primarily of the mathematical background needed for the second part Much of this background is already known and has appeared in various texts Since these results come from many areas of mathematics such as geometry, algebra, topology and logic we thought it convenient to provide a self-contained, coherent exposition of these topics In Chapter and Chapter 2, we study algebraically closed fields (such as the field of complex numbers C) and real closed fields (such as the field of real numbers R) The concept of a real closed field was first introduced by Artin and Schreier in the 1920’s and was used for their solution to Hilbert’s 17th problem [6, 7] The consideration of abstract real closed fields rather than the field of real numbers in the study of algorithms in real algebraic geometry is not only intellectually challenging, it also plays an important role in several complexity results given in the second part of the book CuuDuongThanCong.com Introduction Chapters and describe an interplay between geometry and logic for algebraically closed fields and real closed fields In Chapter 1, the basic geometric objects are constructible sets These are the subsets of Cn which are defined by a finite number of polynomial equations (P = 0) and inequations (P 0) We prove that the projection of a constructible set is constructible The proof is very elementary and uses nothing but a parametric version of the euclidean remainder sequence In Chapter 2, the basic geometric objects are the semi-algebraic sets which constitute our main objects of interest in this book These are the subsets of Rn that are defined by a finite number of polynomial equations (P = 0) and inequalities (P > 0) We prove that the projection of a semi-algebraic set is semi-algebraic The proof, though more complicated than that for the algebraically closed case, is still quite elementary It is based on a parametric version of real root counting techniques developed in the nineteenth century by Sturm, which uses a clever modification of euclidean remainder sequence The geometric statement “the projection of a semi-algebraic set is semi-algebraic” yields, after introducing the necessary terminology, the theorem of Tarski that “the theory of real closed fields admits quantifier elimination.” A consequence of this last result is the decidability of elementary algebra and geometry, which was Tarski’s initial motivation In particular whether there exist real solutions to a finite set of polynomial equations and inequalities is decidable This decidability result is quite striking, given the undecidability result proved by Matijacević [113] for a similar question, Hilbert’s 10-th problem: there is no algorithm deciding whether or not a general system of Diophantine equations has an integer solution In Chapter we develop some elementary properties of semi-algebraic sets Since we work over various real closed fields, and not only over the reals, it is necessary to reexamine several notions whose classical definitions break down in non-archimedean real closed fields Examples of these are connectedness and compactness Our proofs use non-archimedean real closed field extensions, which contain infinitesimal elements and can be described geometrically as germs of semi-algebraic functions, and algebraically as algebraic Puiseux series The real closed field of algebraic Puiseux series plays a key role in the complexity results of Chapters 13 to 16 Chapter describes several algebraic results, relating in various ways properties of univariate and multivariate polynomials to linear algebra, determinants and quadratic forms A general theme is to express some properties of univariate polynomials by the vanishing of specific polynomial expressions in their coefficients The discriminant of a univariate polynomial P , for example, is a polynomial in the coefficients of P which vanishes when P has a multiple root The discriminant is intimately related to real root counting, since, for polynomials of a fixed degree, all of whose roots are distinct, the sign of the discriminant determines the number of real roots modulo The discriminant is in fact the determinant of a symmetric matrix whose signature gives an alternative method to Sturm’s for real root counting due to Hermite CuuDuongThanCong.com ... 143 1-1 550 ISBN-10 3-5 4 0-3 309 8-4 Springer Berlin Heidelberg New York ISBN-13 97 8-3 -5 4 0-3 309 8-1 Springer Berlin Heidelberg New York ISBN 3-5 4 0-0 097 3-6 1st edition Springer-Verlag Berlin Heidelberg... e-mail: Marie-Francoise.Roy@univ-rennes1.fr Library of Congress Control Number: 2006927110 Mathematics Subject Classification (2000): 14P10, 68W30, 03C10, 68Q25, 52C45 ISSN 143 1-1 550 ISBN-10 3-5 4 0-3 309 8-4 ... GA 3033 2-0 160 USA e-mail: saugata@math.gatech.edu Richard Pollack Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 USA e-mail: pollack@cims.nyu.edu Marie-Franỗoise