This book has benefited from several stimulating discussions, especially with M. Anjos, G. Blekherman, R. E. Curto, E. de Klerk, L. A. Fialkow, J. W. Helton, M. Kojima, S. Kuhlmann, M. Laurent, M. Marshall, T. Netzer, J. Nie, P. Parrilo, D. Pasechnich, D. Plaumann, V. Powers, M. Putinar, F. Rendl, B. Reznick, C. Riener, M.-F. Roy, C. Scheiderer, K. Schmüdgen, M. Schweighofer, B. Sturmfels, T. Theobald, K. Toh, and V. Vinnikov, during several visits and/or workshops in particular at the Mathematical Sciences Research Institute (MRSI, Berkeley), the Institute of Pure and Applied Mathematics (IPAM, Los Angeles), the American Institute of Mathematics (AIM, Palo Alto), the Fields Institute for Research in Mathematical Sciences (Toronto), the Institute for Mathematics and its Applications (IMA, Minneapolis), the Oberwolfach Institute, the Isaac Newton Institute (Cambridge, UK), the Institute for Computational and Experimental Research in Mathematics (ICERM, Providence) and the Institute for Mathematical Sciences (IMS, Singapore). I gratefully acknowledge financial support from the above institutes which made these visits possible.

I also want to thank G. Blekherman, F. Bugarin, M. Ghasemi, N. Gravin, R. Laraki, M. Laurent, M. Mevissen, T. Netzer, M. Putinar, S. Robins, P. Rostalski, C. Savorgnan, and T. Phan Thanh, as some of the results presented in this book have been obtained with their collaboration. I would like to thank Cynthia Vinzant who kindly provided the picture on the book cover. Finally, special thanks are also due to my colleague Didier Henrion at LAAS (especially for our collaboration on the software GloptiPoly which he made user friendly with free access), as well as to J.-B. Hiriart Urruty, J. Renegar, S. Sorin, and M. Todd for their constant and friendly support.

For part of this research, financial support from the (French) ANR (GEOLMI project), the Simone and Cino Del Duca foundation (OPTIGACOM project), and the Gaspar Monge Program for Optimization and Operations Research (from the Fondation Mathématique Jacques Hadamard (FMJH)) is gratefully acknowledged.