604 Recent Advances in Real Complexity and Computation UIMP-RSME Lluís A. Santaló Summer School Recent Advances in Real Complexity and Computation July 16–20, 2012 Universidad Internacional Menéndez Pelayo, Santander, Spain José Luis Montaña Luis M. Pardo Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Recent Advances in Real Complexity and Computation UIMP-RSME Lluís A. Santaló Summer School Recent Advances in Real Complexity and Computation July 16–20, 2012 Universidad Internacional Menéndez Pelayo, Santander, Spain José Luis Montaña Luis M. Pardo Editors Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms 604 Recent Advances in Real Complexity and Computation UIMP-RSME Lluís A. Santaló Summer School Recent Advances in Real Complexity and Computation July 16–20, 2012 Universidad Internacional Menéndez Pelayo, Santander, Spain José Luis Montaña Luis M. Pardo Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Providence, Rhode Island Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms EDITORIAL COMMITTEE Dennis DeTurck, managing editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 03D15, 14Qxx, 14Q20, 65-xx, 65H20. Library of Congress Cataloging-in-Publication Data UIMP-RSME Lluis Santal´o Summer School (2012 : Santander, Spain) Recent advances in real complexity and computation : UIMP-RSME Lluis Santal´o Summer School 2012, recent advances in real complexity and computation, July 16–20, 2012, UIMP Palacio de la Magdalena, Santander (Cantabria), Spain / Jose Luis Montana, Luis M. Pardo, editors. pages cm – (Contemporary Mathematics ; volume 604) Includes bibliographical references. ISBN 978-0-8218-9150-6 (alk. paper) 1. Computational complexity–Congresses. I. Pardo, L. M. (Luis M.), editor of compilation. II. Montana, Jose Luis, 1961–editor of compilation. III. Title. QA267.7.U36 2012 511.352–dc23 2013022512 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/604 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. 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License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Dedicated to our beloved friend Jean-Pierre Dedieu. Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents Editors’ preface ix Topics in real and complex number complexity theory Martijn Baartse and Klaus Meer 1 Polar, bipolar and copolar varieties: Real solving of algebraic varieties with intrinsic complexity Bernd Bank, Marc Giusti, and Joos Heintz 55 The complexity and geometry of numerically solving polynomial systems Carlos Beltran´ and Michael Shub 71 Multiplicity hunting and approximating multiple roots of polynomial systems M. Giusti and J.-C. Yakoubsohn 105 On the intrinsic complexity of elimination problems in effective algebraic geometry Joos Heintz, Bart Kuijpers, and Andres´ Rojas Paredes 129 Newton iteration, conditioning and zero counting Gregorio Malajovich 151 vii Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Editors’ preface This volume is composed of six contributions derived from the lectures given during the UIMP–RSME Llu´ıs Santal´o Summer School on “Recent Advances in Real Complexity and Computation”. The goal of this Summer School was to present some of the recent advances on Smale’s 17th Problem. This Problem was stated by Steve Smale as follows: Problem 1 (Smale’s 17th Problem). Can a zero of n complex polynomial equa- tions in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? These contributions cover several aspects around this problem: from numerical to symbolic methods in polynomial equation solving, computational complexity as- pects (both worse and average cases, both upper and lower complexity bounds) and even aspects of the underlying geometry of the problem. Some of the contributions also deal with either real or multiple solutions solving. The School was oriented to graduate mathematicians, as to Master or Ph. D. students in Mathematics and to senior researchers interested on this topic. The School was promoted and supported by the Spanish Royal Mathematical Society (RSME) and hosted by the Universidad Internacional Men´endez Pelayo (UIMP), from July 16th to July 20th of 2012, in El Palacio de la Magdalena, San- tander. Partial financial support was also granted by the University of Cantabria and the Spanish Ministry of Science Grant MTM2010-16051. We thank these in- stitutions and grants for their financial support. The speakers (in alphabetical order) and their courses in this Summer School were the following ones: • Carlos Beltr´an,“Stability, precision and complexity in some numerical problems”. • Marc Giusti, “Polar, co–polar and bipolar varieties: real solving of alge- braic varieties with intrinsic complexity”. • Joos Heintz, “On the intrinsic complexity of elimination problems in ef- fective algebraic geometry”. • Gregorio Malajovich, “From the quadratic convergence of Newton’s method to problems of counting of the number of solutions”. • Klaus Meer,“Real Number Complexity Theory and Probabilistically Check- able Proofs (PCPs)”. • Michael Shub,“The geometry of condition and the analysis of algorithms”. • Jean-Claude Yakoubsohn, “ Tracking multiplicities”. ix Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms x EDITORS’ PREFACE The present volume extends the Summer School by expository articles pre- senting the state of art of each of the topics. The reader will find the following contributions in forthcoming pages: (1) Martijn Baartse and Klaus Meer,“Topics in real and complex num- ber complexity theory” The contribution intends to introduce into topics relevant in real and com- plex number complexity theory. This is done in a survey style. Taking as starting point the computational model introduced by Blum, Shub, and Smale the following issues are addressed: Basic results concerning decid- ability and NP-completeness, transfer results of open questions between different models of computation, structural complexity inside NPR,com- putational