F. Alizadeh, Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization, SIAM Journal on Optimization, vol.5, issue.1, pp.13-51, 1995.
DOI : 10.1137/0805002

R. G. Bartle and D. R. Sherbert, Introduction to real analysis, 1992.

J. Bokowski and B. Sturmfels, Computational synthetic geometry, Lecture Notes in Mathematics, vol.1355, 1989.
DOI : 10.1007/BFb0089253

M. Bucero and B. Mourrain, Exact relaxation for polynomial optimization on semi-algebraic sets, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00846977

A. Alzati, E. Ballico, and G. Ottaviani, The theorem of Mather on generic projections for singular varieties, Geometriae Dedicata, vol.85, issue.1/3, pp.113-117, 2001.
DOI : 10.1023/A:1010383721612

A. Alzati and G. Ottaviani, The theorem of Mather on generic projections in the setting of algebraic geometry, Manuscripta Mathematica, vol.98, issue.1, pp.391-412, 1992.
DOI : 10.1007/BF02567678

C. L. Alonso, J. Puente, and J. L. Montaña, Straight Line Programs: A New Linear Genetic Programming Approach, 2008 20th IEEE International Conference on Tools with Artificial Intelligence, pp.517-524, 2008.
DOI : 10.1109/ICTAI.2008.14

G. Blekherman, Dimensional differences between nonnegative polynomials and sums of squares, 2009.

M. F. Anjos and J. B. , Lasserre editors. Handbook of semidefinite, conic and polynomial optimization. International Series in Operational Research and Management Science, 2012.

E. Arbarello, J. Harris, M. Cornalba, and P. Griffiths, Geometry of algebraic curves. Volume I, Grundlehren der Mathematischen Wissenschaften, vol.267, 1985.

]. B. Bank, M. Giusti, J. Heintz, M. Safey-el-din, and É. Schost, On the geometry of polar varieties, Applicable Algebra in Engineering, Communication and Computing, vol.43, issue.2, pp.33-83, 2010.
DOI : 10.1007/s00200-009-0117-1

URL : https://hal.archives-ouvertes.fr/hal-01148162

B. Bank, M. Giusti, J. Heintz, and G. Mbakop, Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case, Journal of Complexity, vol.13, issue.1, pp.5-27, 1997.
DOI : 10.1006/jcom.1997.0432

B. Bank, M. Giusti, J. Heintz, and G. Mbakop, Polar varieties and efficient real elimination, Mathematische Zeitschrift, vol.238, issue.1, pp.115-144, 2001.
DOI : 10.1007/PL00004896

URL : http://arxiv.org/abs/math/0005041

B. Bank, M. Giusti, J. Heintz, and L. Pardo, Generalized polar varieties and efficient real elimination procedure, Kybernetika, vol.40, issue.5, pp.519-550, 2004.
DOI : 10.1007/pl00004896

URL : http://arxiv.org/abs/math/0005041

B. Bank, M. Giusti, J. Heintz, and L. Pardo, Generalized polar varieties: geometry and algorithms, Journal of Complexity, vol.21, issue.4, pp.377-412, 2005.
DOI : 10.1016/j.jco.2004.10.001

URL : http://doi.org/10.1016/j.jco.2004.10.001

B. Bank, M. Giusti, J. Heintz, and L. Pardo, Bipolar varieties and real solving of a singular polynomial equation, Jaen Journal of Approximation, vol.2, issue.1, pp.65-77, 2010.

A. Bhardwaj, P. Rostalski, and R. Sanyal, Deciding Polyhedrality of Spectrahedra, SIAM Journal on Optimization, vol.25, issue.3, 2015.
DOI : 10.1137/120904172

S. Basu, R. Pollack, and M. Roy, On the number of cells defined by a family of polynomials on a variety Mathematika, pp.120-126, 1996.

S. Basu, R. Pollack, and M. Roy, On the Betti numbers of sign conditions Proceedings of the, pp.965-974, 2004.

S. Basu, R. Pollack, and M. Roy, Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, vol.10, 2006.
DOI : 10.1007/978-3-662-05355-3

URL : https://hal.archives-ouvertes.fr/hal-01083587

A. Beauville, Determinantal hypersurfaces., The Michigan Mathematical Journal, vol.48, issue.1, pp.39-64, 2000.
DOI : 10.1307/mmj/1030132707

A. Ben-tal and A. Nemirovski, Lectures on modern convex optimization: analysis, algorithms, engineering applications, MPS-SIAM Series on Optimization, 2001.
DOI : 10.1137/1.9780898718829

G. Blekherman, P. A. Parrilo, and R. R. Thomas, Semidefinite optimization and convex algebraic geometry, 2013.
DOI : 10.1137/1.9781611972290

J. Bochnak, M. Coste, and M. Roy, Real algebraic geometry, 1998.
DOI : 10.1007/978-3-662-03718-8

S. P. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory, Studies in Applied Mathematics SIAM, vol.15, 1994.
DOI : 10.1137/1.9781611970777

L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, vol.38, issue.1, pp.49-95, 1996.
DOI : 10.1137/1038003

W. Bruns and U. Vetter, Determinantal rings, 1988.
DOI : 10.1017/cbo9780511608681.009

B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, ACM SIGSAM Bulletin, vol.10, issue.3, pp.19-29, 1976.
DOI : 10.1145/1088216.1088219

P. Bürgisser, M. Clausen, and M. A. Shokrollahi, Algebraic complexity theory. Grundlehren der mathematischen Wissenschaften 315, 1997.

M. D. Choi, T. Y. Lam, and B. Reznick, Sums of squares of real polynomials, Proceedings of Symposia in Pure mathematics, pp.103-126, 1995.
DOI : 10.1090/pspum/058.2/1327293

M. Claeys, Mesures d'occupation et relaxations semi-définies pour la commande optimale, LAAS CNRS, 2013.

G. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion. Automata Theory and Formal Languages, pp.134-183, 1975.

A. Conca, Straightening Law and Powers of Determinantal Ideals of Hankel Matrices, Advances in Mathematics, vol.138, issue.2, pp.263-292, 1998.
DOI : 10.1006/aima.1998.1740

D. A. Cox, J. Little, and D. Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, 2007.

J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels, and R. R. Thomas, The Euclidean Distance Degree of an Algebraic Variety, Foundations of Computational Mathematics, vol.70, issue.2, 2015.
DOI : 10.1007/s10208-014-9240-x

C. Durvye, Algorithmes pour la décomposition primaire des idéaus polynomiaux de dimension nulle donné en évaluation, 2008.

D. Eisenbud, Linear Sections of Determinantal Varieties, American Journal of Mathematics, vol.110, issue.3, pp.541-575, 1988.
DOI : 10.2307/2374622

J. Faugère, A new efficient algorithm for computing Gröbner bases (F4), Journal of Pure and Applied Algebra, vol.139, pp.1-361, 1999.

J. Faugère, A new efficient algorithm for computing Gröbner bases without reductions to zero (F5), Proceedings of ISSAC 2002, 2002.

J. Faugère, FGb: A Library for Computing Gr??bner Bases, Mathematical Software?ICMS 2010, pp.84-87, 2010.
DOI : 10.1007/978-3-642-15582-6_17

J. Faugère, M. Safey-el-din, and P. Spaenlehauer, On the complexity of the generalized MinRank problem, Journal of Symbolic Computation, vol.55, pp.30-58, 2013.
DOI : 10.1016/j.jsc.2013.03.004

J. Faugère, M. Safey-el-din, and P. Spaenlehauer, Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology, Proceedings of ISSAC 2010, 2010.

J. Faugère, F. Levy-dit-vehel, and L. Perret, Cryptanalysis of MinRank, Lecture Notes in Computer Science, vol.5157, pp.280-296, 2008.
DOI : 10.1007/978-3-540-85174-5_16

J. Faugère, P. Gaudry, L. Huot, and G. Renault, Polynomial systems solving by fast linear algebra, 2013.

J. Faugère, P. Gianni, D. Lazard, and T. Mora, Efficient Computation of Zero-dimensional Gr??bner Bases by Change of Ordering, Journal of Symbolic Computation, vol.16, issue.4, pp.329-344, 1993.
DOI : 10.1006/jsco.1993.1051

J. Faugère and C. Mou, Sparse FGLM algorithms, Journal of Symbolic Computation, vol.80, 2013.
DOI : 10.1016/j.jsc.2016.07.025

J. Faugère and C. Mou, Fast algorithm for change of ordering of zerodimensional Gröbner bases with sparse multiplication matrices, Proceedings of ISSAC 2011, 2011.

I. M. Gel-'fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Mathematics: Theory & Applications, 1994.

M. Giusti, K. Hägele, G. Lecerf, J. Marchand, and B. Salvy, The Projective Noether Maple Package: Computing the Dimension of a Projective Variety, Journal of Symbolic Computation, vol.30, issue.3, pp.291-307, 2000.
DOI : 10.1006/jsco.2000.0369

URL : https://hal.archives-ouvertes.fr/inria-00073465

M. Giusti, G. Lecerf, and B. Salvy, A Gr??bner Free Alternative for Polynomial System Solving, Journal of Complexity, vol.17, issue.1, pp.154-211, 2001.
DOI : 10.1006/jcom.2000.0571

H. G. Bothmer and K. Ranestad, A general formula for the algebraic degree in semidefinite programming Bulletin of LMS, pp.193-197, 2009.

D. Grigoriev and N. Vorobjov, Solving systems of polynomial inequalities in subexponential time, Journal of Symbolic Computation, vol.5, issue.1-2, pp.37-64, 1988.
DOI : 10.1016/S0747-7171(88)80005-1

D. Grigoriev and D. Pasechnik, Polynomial-time computing over quadratic maps i: sampling in real algebraic sets, computational complexity, vol.14, issue.1, pp.20-52, 2005.
DOI : 10.1007/s00037-005-0189-7

URL : https://hal.archives-ouvertes.fr/hal-00725128

A. Greuet, Polynomial optimization and polar varieties: theory, algorithms and implementations, 2013.
URL : https://hal.archives-ouvertes.fr/tel-00922805

A. Greuet and M. , Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set, SIAM Journal on Optimization, vol.24, issue.3, pp.1313-1343, 2014.
DOI : 10.1137/130931308

URL : https://hal.archives-ouvertes.fr/hal-00849523

M. Grötschel, L. Lovász, and A. Schrijver, Geometric algorithms and combinatorial optimization, 1988.

Q. Guo, M. Safey-el-din, and L. Zhi, Computing rational solutions of linear matrix inequalities, Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation, ISSAC '13
DOI : 10.1145/2465506.2465949

URL : https://hal.archives-ouvertes.fr/hal-00815174

F. Guo, E. Kaltofen, and L. Zhi, Certificates of impossibility of Hilbert-Artin representations of a given degree for definite polynomials and functions, Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ISSAC '12, pp.195-202, 2012.
DOI : 10.1145/2442829.2442859

J. Harris, Algebraic geometry. A first course, 1992.

R. Hartshorne, Algebraic geometry, 1977.
DOI : 10.1007/978-1-4757-3849-0

G. Heinig and K. Rost, Algebraic methods for Toeplitz-like matrices and operators, 1984.
DOI : 10.1007/978-3-0348-6241-7

J. Heintz, M. Roy, and P. Solerno, On the Theoretical and Practical Complexity of the Existential Theory of Reals, The Computer Journal, vol.36, issue.5, pp.427-431, 1993.
DOI : 10.1093/comjnl/36.5.427

J. Heintz, M. Roy, and P. Solerno, Description of the connected components of a semialgebraic set in single exponential time, Discrete & Computational Geometry, vol.4, issue.4, pp.121-140, 1994.
DOI : 10.1007/BF02573999

J. Heintz, M. Roy, and P. Solerno, On the complexity of semi-algebraic sets, Proc. IFIP 89, pp.293-298, 1989.

D. Henrion, S. Naldi, and M. , Safey El Din Real root finding for determinants of linear matrices, Journal of Symbolic Computation, 2015.

]. D. Henrion, S. Naldi, and M. , Safey El Din. Real root finding for rank defects in linear Hankel matrices, Proceedings of ISSAC 2015, 2015.

D. Henrion, S. Naldi, and M. , Safey El Din Real root finding for low rank linear matrices. To be registered as a LAAS-CNRS Research Report, 2015.

D. Henrion, S. Naldi, and M. , Safey El Din Exact algorithms for linear matrix inequalities, 2015.

B. Helton and J. Nie, Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets, SIAM Journal on Optimization, vol.20, issue.2, pp.759-791, 2009.
DOI : 10.1137/07070526X

D. Henrion, J. B. Lasserre, and J. Löfberg, GloptiPoly 3: moments, optimization and semidefinite programming, Optimization Methods and Software, vol.24, issue.4-5, pp.4-5761, 2009.
DOI : 10.1080/10556780802699201

URL : https://hal.archives-ouvertes.fr/hal-00172442

B. Huber and B. Sturmfels, A polyhedral method for solving sparse polynomial systems, Mathematics of Computation, vol.64, issue.212, pp.1541-1555, 1995.
DOI : 10.1090/S0025-5718-1995-1297471-4

J. B. Lasserre, D. Henrion, C. Prieur, and E. Trélat, Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations, SIAM Journal on Control and Optimization, vol.47, issue.4, pp.1643-1666, 2008.
DOI : 10.1137/070685051

URL : https://hal.archives-ouvertes.fr/hal-00136032

D. Henrion, J. B. Lasserre, and C. Savorgnan, Approximate Volume and Integration for Basic Semialgebraic Sets, SIAM Review, vol.51, issue.4, pp.722-743, 2009.
DOI : 10.1137/080730287

URL : https://hal.archives-ouvertes.fr/hal-00297384

D. Henrion, Semidefinite geometry of the numerical range, Electronic Journal of Linear Algebra, vol.20, issue.1, pp.322-332, 2010.
DOI : 10.13001/1081-3810.1377

URL : https://hal.archives-ouvertes.fr/hal-00345031

D. Henrion, Optimization on linear matrix inequalities for polynomial systems control, Lecture notes of the International Summer School of Automatic Control, 2013.
DOI : 10.5802/ccirm.17

URL : https://hal.archives-ouvertes.fr/hal-00860856

D. Henrion and J. Malick, Projection methods for conic feasibility problems: applications to polynomial sum-of-squares decompositions, Optimization Methods and Software, vol.26, issue.1, pp.23-46, 2011.
DOI : 10.1080/10556780903191165

V. Magron, D. Henrion, and J. B. Lasserre, Semidefinite approximations of projections and polynomial images of semialgebraic sets To be registered as a LAAS-CNRS Research Report, 2014.

D. Hilbert, Ueber die Theorie der algebraischen Formen, Mathematische Annalen, vol.36, issue.4, pp.473-534, 1890.
DOI : 10.1007/BF01208503

D. Hilbert, Ueber die vollen Invariantensysteme, Mathematische Annalen, vol.42, issue.3, pp.13-373, 1893.
DOI : 10.1007/BF01444162

D. Hilbert, Ueber die Darstellung definiter Formen als Summe von Formenquadraten, Mathematische Annalen, vol.32, issue.3, pp.342-350, 1888.
DOI : 10.1007/BF01443605

R. Hildebrand, Spectrahedral cones generated by rank 1 matrices, Journal of Global Optimization, vol.48, issue.3
DOI : 10.1007/s10898-015-0313-4

URL : https://hal.archives-ouvertes.fr/hal-01417475

C. Hillar, Sums of polynomial squares over totally real fields are rational sums of squares, Proc. American Math. Society, pp.921-930, 2009.

J. Huh and B. Sturmfels, Likelihood Geometry, Combinatorial Algebraic Geometry, pp.63-117, 2014.
DOI : 10.1007/978-3-319-04870-3_3

Z. Jelonek, Testing sets for properness of polynomial mappings, Mathematische Annalen, vol.315, issue.1, pp.1-35, 1999.
DOI : 10.1007/s002080050316

G. Jeronimo, G. Matera, P. Solernó, and A. Waissbein, Deformation Techniques for Sparse Systems, Foundations of Computational Mathematics, vol.31, issue.3, pp.1-50, 2009.
DOI : 10.1007/s10208-008-9024-2

G. Jeronimo, D. Perrucci, and J. Sabia, On Sign Conditions Over Real Multivariate Polynomials, Discrete & Computational Geometry, vol.60, issue.5, pp.195-222, 2010.
DOI : 10.1007/s00454-009-9200-4

D. Jibetean and E. De-klerk, Global optimization of rational functions: a semidefinite programming approach, Mathematical Programming, pp.93-109, 2006.
DOI : 10.1007/s10107-005-0589-0

L. Khachiyan and L. Porkolab, On the complexity of semidefinite programs, J. Global Optim, vol.10, pp.351-365, 1997.

H. Khalil, Nonlinear systems, 2002.

J. Kollar, Sharp Effective Nullstellensatz, Journal of the American Mathematical Society, vol.1, issue.4, pp.963-975, 1988.
DOI : 10.2307/1990996

I. Klep and M. Schweighofer, An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares, Mathematics of Operations Research, vol.38, issue.3, pp.569-590, 2013.
DOI : 10.1287/moor.1120.0584

I. Klep and M. Schweighofer, Pure states, positive matrix polynomials and sums of hermitian squares, Indiana University Mathematics Journal, vol.59, issue.3, pp.857-874, 2010.
DOI : 10.1512/iumj.2010.59.4107

I. Klep and M. Schweighofer, Infeasibility certificates for linear matrix inequalities, Oberwolfach Preprints (OWP), vol.28, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00617640

F. and L. Gall, Powers of tensors and fast matrix multiplication, Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ISSAC '14, pp.296-303, 2002.
DOI : 10.1145/2608628.2608664

J. B. Lasserre, Global Optimization with Polynomials and the Problem of Moments, SIAM Journal on Optimization, vol.11, issue.3, pp.796-817, 2001.
DOI : 10.1137/S1052623400366802

J. B. Lasserre, A semidefinite programming approach to the generalized problem of moments, Mathematical Programming, pp.65-92, 2008.
DOI : 10.1007/s10107-006-0085-1

M. Laurent, Sums of squares, moment matrices and optimization over polynomials Emerging Applications of Algebraic Geometry, of IMA Volumes in Mathematics and its Applications, pp.157-270, 2009.

C. , L. Guernic, F. Rouillier, and M. , Safey El Din On the practical computation of one point in each connected component of a semi-algebraic set defined by a polynomial system of equations and non-strict inequalities, Proceedings of EACA 04 conference, L.Gonzalez-Vega and T. Recio Eds, 2004.

A. Logar, A computational proof of the Noether normalization lemma, Lecture Notes in Computer Science, vol.357, pp.259-273, 1989.
DOI : 10.1007/3-540-51083-4_65

P. Gahinet and A. Nemirovsky, LMI Lab: A package for manipulating and solving LMIs, 1993.

H. Lombardi, D. Perrucci, and M. Roy, An elementary recursive bound for the effective Positivestellensatz and Hilbert 17th problem, 2013.

D. R. Grayson and M. E. Stillman, macaulay2, a software system for research in algebraic geometry: http://www

J. N. Mather, Generic Projections, The Annals of Mathematics, vol.98, issue.2, pp.226-245, 1973.
DOI : 10.2307/1970783

Y. Ma and L. Zhi, Computing real solutions of polynomial systems via lowrank moment matrix completion, Proceedings of ISSAC 2012, pp.249-256, 2012.

S. Naldi, Nonnegative Polynomials and Their Carath??odory Number, Discrete & Computational Geometry, vol.57, issue.2, pp.559-568, 2014.
DOI : 10.1007/s00454-014-9588-3

URL : http://arxiv.org/abs/1209.3298

Y. Nesterov and A. Nemirovsky, Interior-point polynomial algorithms in convex programming, Studies in Applied Mathematics, vol.13, 1994.
DOI : 10.1137/1.9781611970791

Y. Nesterov, Squared functional systems and optimization problems High performance optimization, pp.405-440, 2000.

J. Nie, K. Ranestad, and B. Sturmfels, The algebraic degree of semidefinite programming, Mathematical Programming, vol.296, issue.12, pp.379-405, 2010.
DOI : 10.1007/s10107-008-0253-6

J. Nie, Optimality conditions and finite convergence of Lasserre???s hierarchy, Mathematical Programming, vol.352, issue.2, pp.97-121, 2014.
DOI : 10.1007/s10107-013-0680-x

J. Nie and M. Schweighofer, On the complexity of Putinar's Positivstellensatz, Journal of Complexity, vol.23, issue.1, pp.135-150, 2007.
DOI : 10.1016/j.jco.2006.07.002

G. Ottaviani, P. Spaenlehauer, and B. Sturmfels, Exact Solutions in Structured Low-Rank Approximation, SIAM Journal on Matrix Analysis and Applications, vol.35, issue.4, pp.1521-1542, 2014.
DOI : 10.1137/13094520X

URL : https://hal.archives-ouvertes.fr/hal-00953702

P. Parrilo, Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, vol.96, issue.2, pp.293-320, 2003.
DOI : 10.1007/s10107-003-0387-5

P. Parrilo and B. Sturmfels, Minimizing polynomial functions Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in, Mathematics and Computer Science, pp.83-100, 2001.

V. Y. Pan, E. Tsigaridas, and Z. Liang, Simple and efficient real root finding for a univariate polynomial
URL : https://hal.archives-ouvertes.fr/hal-01105309

G. Pataki, The geometry of cone-LP's, Saigal, Vandenberghe Eds. Handbook of Semidefinite Programming, pp.29-66, 2000.

D. Plaumann, B. Sturmfels, and C. Vinzant, Quartic curves and their bitangents, Journal of Symbolic Computation, vol.46, issue.6, pp.712-733, 2011.
DOI : 10.1016/j.jsc.2011.01.007

URL : http://doi.org/10.1016/j.jsc.2011.01.007

J. C. Ottem, K. Ranestad, B. Sturmfels, and C. Vinzant, Quartic spectrahedra, Mathematical Programming, Series B, Special issue on Polynomial Optimization, 2013.
DOI : 10.1007/s10107-014-0844-3

URL : https://www.duo.uio.no/bitstream/10852/41852/2/qs.pdf

M. Ramana and A. J. Goldman, Some geometric results in semidefinite programming, Journal of Global Optimization, vol.40, issue.2, pp.33-50, 1995.
DOI : 10.1007/BF01100204

M. Ramana, L. Tunçel, and H. Wolkowicz, Strong Duality for Semidefinite Programming, SIAM Journal on Optimization, vol.7, issue.3, pp.641-662, 1997.
DOI : 10.1137/S1052623495288350

J. Renegar, On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals, Journal of Symbolic Computation, vol.13, issue.3, pp.255-352, 1992.
DOI : 10.1016/S0747-7171(10)80003-3

B. Reznick, Sums of even powers of real linear forms, Memoirs of the American Mathematical Society, vol.96, issue.463, 1992.
DOI : 10.1090/memo/0463

M. Safey-el-din, Raglib (real algebraic geometry library), Maple package

F. Rouillier, Solving Zero-Dimensional Systems Through the Rational Univariate Representation, Applicable Algebra in Engineering, Communication and Computing, vol.9, issue.5, pp.433-461, 1999.
DOI : 10.1007/s002000050114

URL : https://hal.archives-ouvertes.fr/inria-00073264

M. Safey-el-din and É. Schost, Polar varieties and computation of one point in each connected component of a smooth real algebraic set, Proceedings of the 2003 international symposium on Symbolic and algebraic computation , ISSAC '03, pp.224-231, 2003.
DOI : 10.1145/860854.860901

URL : https://hal.archives-ouvertes.fr/inria-00099649

M. Safey-el-din and É. Schost, Properness defects of projections and computation of one point in each connected component of a real algebraic set, Discrete and Computational Geometry, vol.32, issue.3, pp.417-430, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00099962

M. Safey-el-din and É. Schost, A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00849057

M. Safey-el-din, Finding sampling points on real hypersurfaces is easier in singular situations, Electronic proceedings of MEGA (Effective Methods in Algebraic Geometry), 2005.

R. Sanyal, F. Sottile, and B. , ORBITOPES, Mathematika, vol.22, issue.02, pp.275-314, 2011.
DOI : 10.1007/BF01443605

C. Scheiderer, Sums of squares of polynomials with rational coefficients, Journal of the European Mathematical Society, vol.18, issue.7, 2013.
DOI : 10.4171/JEMS/620

C. Scheiderer, Semidefinite representation for convex hulls of real algebraic curves, 2012.

C. Scheiderer, Positivity and Sums of Squares: A Guide to Recent Results, Emerging Applications of Algebraic Geometry, pp.271-324, 2009.
DOI : 10.1007/978-0-387-09686-5_8

I. Shafarevich, Basic algebraic geometry 1, 1977.
DOI : 10.1007/978-3-642-57908-0

K. Schmüdgen, TheK-moment problem for compact semi-algebraic sets, Mathematische Annalen, vol.207, issue.1, pp.203-206, 1991.
DOI : 10.1007/BF01446568

M. Schweighofer, On the complexity of Schm??dgen's Positivstellensatz, Journal of Complexity, vol.20, issue.4, pp.529-543, 2004.
DOI : 10.1016/j.jco.2004.01.005

R. Sinn and B. Sturmfels, Generic Spectrahedral Shadows, SIAM Journal on Optimization, vol.25, issue.2, pp.1209-1220, 2015.
DOI : 10.1137/140978478

URL : http://arxiv.org/abs/1407.5219

P. Spaenlehauer, Résolution de systèmes multi-homogènes et déterminantiels, 2012.

M. Spivak, Calculus on manifolds, WA Benjamin New York, vol.1, 1965.

V. Strassen, Gaussian elimination is not optimal, Numerische Mathematik, vol.13, issue.4, pp.354-356, 1969.
DOI : 10.1007/BF02165411

URL : http://www.digizeitschriften.de/download/PPN362160546_0013/PPN362160546_0013___log38.pdf

J. F. Sturm, SeDuMi version 1, 2006.

B. Sturmfels, What is a Gröbner basis. Notices of the, pp.1199-1200, 2005.

F. Tanturri, Degeneracy loci of morphisms between vector bundles

S. Tarbouriech, G. Garcia, J. M. Gomes-da-silva, and I. Queinnec, Stability and stabilization of linear systems with saturating actuators, 2011.
DOI : 10.1007/978-0-85729-941-3

A. Tarski, A Decision Method for Elementary Algebra and Geometry, 1951.
DOI : 10.1007/978-3-7091-9459-1_3

M. J. Todd, Semidefinite optimization, Acta Numerica, vol.10, pp.515-560, 2001.
DOI : 10.1017/S0962492901000071

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.5569

J. Löfberg, YALMIP : a toolbox for modeling and optimization in MATLAB, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), 2004.
DOI : 10.1109/CACSD.2004.1393890

A. Varvitsiotis, Combinatorial conditions for low rank solutions in semidefinite programming, 2013.

C. Vinzant, Real algebraic geometry in convex optimization, 2011.