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Real Algebraic Geometry Saugata Basu, Johannes Huisman, Kurdyka Krzysztof, Victoria Powers, Jean-Philippe Rolin Real Algebraic Geometry Saugata Basu, Johannes Huisman, Kurdyka Krzysztof, Victoria Powers, Jean-Philippe Rolin To cite this version: Saugata Basu, Johannes Huisman, Kurdyka Krzysztof, Victoria Powers, Jean-Philippe Rolin. Real Al- gebraic Geometry. Real Algberaic Geometry, Rennes, 2011, Jun 2011, Rennes, France. hal-00609687 HAL Id: hal-00609687 https://hal.archives-ouvertes.fr/hal-00609687 Submitted on 19 Jul 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Conference Real Algebraic Geometry Université de Rennes 1 20 - 24 june 2011 Beaulieu Campus Conference Real Algebraic Geometry Université de Rennes 1 20 - 24 june 2011 Beaulieu Campus Conference Real Algebraic Geometry C’est un très grand plaisir pour nous d’accueillir à Rennes cette conférence internationale de Géométrie Algébrique Réelle qui est le quatrième opus d’une belle série commencée il y a 30 ans. La toute première conférence internationale de la discipline fut organisée à Rennes en 1981 par Jean-Louis Colliot-Thélène, Michel Coste, Louis Mahé et Marie-Françoise Roy. Depuis, la discipline a pris un essor considérable : de très nombreuses conférences sur la thématique se sont déroulées dans le monde entier, un réseau européen a tissé des liens au cœur duquel Rennes 1 a toujours joué un rôle privilégié. Le laboratoire de géométrie algébrique réelle rennais a acquis au fil des années une notoriété internationale et a poursuivi son travail de fond en organisant deux nouvelles conférences internationales de synthèse en 1991 puis en 2001. Nous sommes donc particulièrement heureux de perpétuer cette tradition décennale d’excellence en ce mois de Juin 2011. Nous voyons aussi en cette conférence l’occasion de rendre un hommage chaleureux aux professeurs Michel Coste, Louis Mahé et Marie-Françoise Roy pour la mise sur pieds et l’animation de l’équipe rennaise de recherche en Géométrie Algébrique Réelle ainsi que pour leur contribution scientifique marquante au renouveau de la thématique éponyme. A l’avant veille — ou au lendemain matin pour l’un d’entre eux qui a pris un peu d’avance ! — de prendre un peu de distance avec leur fonction de mathématicien en exercice à Rennes 1, qu’ils reçoivent tout trois notre plus profonde considération. Nous remercions les 150 chercheurs en mathématiques venus du monde entier pour assister à la conférence et notamment les participants qui vont y donner un exposé — grand merci au passage au comité scientifique. Notre gratitude va tout particulièrement aux conférenciers qui ont accepté le travail ingrat — mais ô combien utile pour la communauté — de rédiger pour la conférence un survol de leur discipline de prédilection : Saugata Basu, Johannes Huisman, Krzyztof Kurdyka, Victoria Powers et Jean-Philippe Rolin. Nous souhaitons aussi remercier les nombreux sponsors qui nous ont permis d’accueillir autant de participants à des conditions avantageuses et notamment un aussi grand nombre de doctorants et post-doctorants appelés à constituer le vivier de la Géométrie Algébrique Réelle de demain. Merci donc à : L’IRMAR, l’Université de Rennes 1, le CNRS, Rennes Métropole, le Conseil Régional, le Ministère de l’enseignement supérieur et de la recherche, Et aussi tout particulièrement à : La fondation Métivier, le GDR CNRS 2945 «Singularités et Applications», les deux supports IUF de Dominique Cerveau et d’Antoine Chambert-Loir, les ANR SIRE et ANR SIROPA. Enfin, nous sommes fortement redevables envers Chantal Halet et Emmanuelle Guiot pour le travail considérable qu’elles ont déployé pour la bonne tenue de cette conférence. Karim Bekka, Goulwen Fichou, Jean-Philippe Monnier, Ronan Quarez Organisateurs de la conférence RAG2011, Le 20 Juin 2011 5 Conference Real Algebraic Geometry It is a very big pleasure for us to welcome in Rennes this international conference in Real Algebraic Geometry which is the fourth opus of a beautiful series begun 30 years ago. The very first international conference of the discipline was organized in Rennes in 1981 by Jean-Louis Colliot-Thélène, Michel Coste, Louis Mahé and Marie-Françoise Roy. Since then, the discipline took a considerable development: very numerous conferences on the theme took place all over the world, a European network forged ties the heart of which Rennes 1 always played a privileged role. Over the years, the Rennes’s real algebraic geometry laboratory acquired an international reputation and continued its thorough work by organizing two new international conferences of synthesis in 1991 then in 2001. We are thus particularly happy to immortalize this ten-year tradition of excellence in this June 2011. We also see in this conference the opportunity to pay a warm tribute to professors Michel Coste, Louis Mahé and Marie-Françoise Roy for the setting up and the animation of the Rennes’s team of research in Real Algebraic Geometry as well as for their striking scientific contribution to the revival of the eponymic theme. On the eve — or the following day morning for one of them who set a little beforehand ! — of setting a little outstrips with their position of mathematician in Rennes 1, may all three receive our deepest consideration. We thank the 150 researchers in mathematics coming from all over world for attending the conference, in particular the participants who are going to give a talk — big thanks to the scientific committee. Our gratitude goes quite particularly to the speakers who accepted the thankless work - but how so useful for the community - to write for the conference a survey of their prefered discipline: Saugata Basu, Johannes Huisman, Krzyztof Kurdyka, Victoria Powers and Jean-Philippe Rolin. We also wish to thank the numerous sponsors who allowed us to welcome so many participants under proper conditions, in particular a large number of PhD students and post-PhD students called to constitute the pool of the Real Algebraic Geometry of tomorrow. Thank you thus to: The IRMAR, the university of Rennes 1, the CNRS, the Rennes Métropole, the Regional Council, the Ministry for Higher Education and Research, and particularly to: The foundation Métivier, the GDR CNRS 2945 «Singularités et Applications», the two supports of the IUF of Dominique Cerveau and of Antoine Chambert-Loir, the ANR SIRE and ANR SIROPA. Finally, we are deeply indebted to Chantal Halet and Emmanuelle Guiot for the considerable work they have done for the good running of this conference. Karim Bekka, Goulwen Fichou, Jean-Philippe Monnier, Ronan Quarez Organizers of the conference RAG2011, June 20th, 2011 7 Conference Real Algebraic Geometry CONTENTS Five surveys on Real Algebraic Geometry.............................................................................P. 11 Algorithms in real algebraic geometry: a survey Saugata Basu............................................................................................................ ...P. 13 Topology of real algebraic varieties; some recent results on rational surfaces Johannes Huisman.......................................................................................................P. 51 Analytic arcs end real analytic singularities Krzysztof Kurdyka.........................................................................................................P. 63 Positive Polynomials and Sums of Squares: Theory and Practice Victoria Powers.............................................................................................................P. 77 A survey on o-minimal structures Jean-Philippe Rolin.......................................................................................................P. 105 Abstracts............................................................................................................................. ..P. 121 Participants............................................................................................................................P. 141 9 FIVE SURVEYS ON REAL ALGEBRAIC GEOMETRY Conference Real Algebraic Geometry ALGORITHMS IN REAL ALGEBRAIC GEOMETRY: A SURVEY SAUGATA BASU Abstract. We survey both old and new developments in the theory of al- gorithms in real algebraic geometry – starting from effective quantifier elim- ination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for computing topological invariants of semi-algebraic sets. We emphasize throughout the complexity aspects of these algorithms and also discuss the computational hardness of the underlying problems. We also describe some recent results linking the computational hardness of decision problems in the first order theory of the reals, with that of computing certain topological invariants of semi-algebraic sets. Even though we mostly concen- trate on exact algorithms, we also discuss some numerical approaches involving semi-definite programming that have gained popularity in recent times. 1. Introduction We survey developments in the theory of algorithms in real algebraic geometry – starting from the first effective quantifier elimination procedure due to Tarski and Seidenberg, to more recent work on efficient
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