M. Bucero and B. Mourrain, Certified relaxation for polynomial optimization on semi-algebraic sets

M. Bucero, B. Mourrain, and P. Trebuchet, Unconstraint global polynomial optimization via Gradient Ideal
URL : https://hal.archives-ouvertes.fr/hal-00779666

C. Aholt, S. Agarwal, and R. Thomas, A QCQP Approach to Triangulation, Computer Vision?ECCV 2012, pp.654-667, 2012.
DOI : 10.1007/978-3-642-33718-5_47

C. Aholt, B. Sturmfels, and R. Thomas, A Hilbert Scheme in Computer Vision, Journal canadien de math??matiques, vol.65, issue.5, 2011.
DOI : 10.4153/CJM-2012-023-2

D. S. Arnon, G. E. Collins, and S. Mccallum, Cylindrical Algebraic Decomposition I: The Basic Algorithm, SIAM Journal on Computing, vol.13, issue.4, pp.865-877, 1984.
DOI : 10.1137/0213054

E. Artin, Über die zerlegung definiter funktionen in quadrate, pp.100-115, 1927.

M. Atiyah and I. Macdonald, Introduction to commutative algebra, 1969.

B. Balasundaram and S. Butenko, Constructing test functions for global optimization using continuous formulations of graph problems, Optimization Methods and Software, vol.3, issue.4-5, pp.439-452, 2005.
DOI : 10.1023/A:1017969603632

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

B. Bank, M. Giusti, J. Heintz, and L. M. 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

B. Bank, M. Giusti, J. Heintz, and M. , Intrinsic complexity estimates in polynomial optimization, Journal of Complexity, vol.30, issue.4, 2013.
DOI : 10.1016/j.jco.2014.02.005

URL : https://hal.archives-ouvertes.fr/hal-00815123

B. Bank, M. Giusti, J. Heintz, M. Safey-el-din, and E. 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

S. Basu, R. Pollack, and M. Roy, On the combinatorial and algebraic complexity of quantifier elimination, Journal of the ACM, vol.43, issue.6, pp.1002-1045, 1996.
DOI : 10.1145/235809.235813

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

W. Baur and V. Strassen, The complexity of partial derivatives, Theoretical Computer Science, vol.22, issue.3, pp.317-330, 1983.
DOI : 10.1016/0304-3975(83)90110-X

R. Berr and T. Wörmann, Positive polynomials and tame preorderings, Mathematische Zeitschrift, vol.236, issue.4, pp.813-840, 2001.
DOI : 10.1007/PL00004853

G. Blekherman, There are significantly more nonegative polynomials than sums of squares, Israel Journal of Mathematics, vol.253, issue.2, pp.355-380, 2006.
DOI : 10.1007/BF02771790

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

S. Boyd and L. Vandenberghe, Convex Optimization, 2004.

C. W. Brown, Solution formula construction for truth-invariant cads, 1999.

C. W. Brown, QEPCAD B, ACM SIGSAM Bulletin, vol.37, issue.4, pp.97-108, 2003.
DOI : 10.1145/968708.968710

W. Bruns and R. Schwänzl, The Number of Equations Defining a Determinantal Variety, Bulletin of the London Mathematical Society, vol.22, issue.5, pp.439-445, 1990.
DOI : 10.1112/blms/22.5.439

W. Bruns and U. Vetter, Determinantal rings, 1988.

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

G. Cassier, Probl??me des moments sur un compact de Rn et d??composition de polyn??mes a plusieurs variables, Journal of Functional Analysis, vol.58, issue.3, pp.254-266, 1984.
DOI : 10.1016/0022-1236(84)90042-9

C. Chen, M. Moreno-maza, B. Xia, and L. Yang, Computing cylindrical algebraic decomposition via triangular decomposition, Proceedings of the 2009 international symposium on Symbolic and algebraic computation, ISSAC '09, pp.95-102, 2009.
DOI : 10.1145/1576702.1576718

M. D. Choi, T. Y. Lam, and B. Reznick, Sums of squares of real polynomials. In K-theory and algebraic geometry: connections with quadratic forms and division algebras, of Proc. Sympos. Pure Math, pp.103-126, 1992.

T. F. Coleman and A. P. Liao, An efficient trust region method for unconstrained discrete-time optimal control problems, Computational Optimization and Applications, vol.15, issue.1, pp.47-66, 1995.
DOI : 10.1007/BF01299158

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Lecture Notes in Comput. Sci, vol.33, pp.134-183, 1975.
DOI : 10.1007/3-540-07407-4_17

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Lecture Notes in Comput. Sci, vol.33, pp.134-183, 1975.
DOI : 10.1007/3-540-07407-4_17

G. E. Collins and H. Hong, Partial cylindrical algebraic decomposition for quantifier elimination, Quantifier elimination and cylindrical algebraic decomposition, pp.174-200, 1993.

C. W. Commander, Maximum Cut Problem, MAX-CUT, Encyclopedia of Optimization, pp.1991-1999, 2009.
DOI : 10.1007/978-0-387-74759-0_358

M. Coste, An Introduction to Semialgebraic Geometry Dottorato di ricerca in matematica, 2000.

P. Cousot, Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Relaxation and Semidefinite Programming, Model Checking, and Abstract Interpretation, pp.1-24, 2005.
DOI : 10.1007/978-3-540-30579-8_1

D. Cox, J. Little, and D. Shea, Ideals, Varieties and Algorithms, 2006.

E. De-klerk, Aspects of Semidefinite Programming: interior point algorithms and selected applications, 2002.
DOI : 10.1007/b105286

J. Demmel, J. Nie, and V. Powers, Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals, Journal of Pure and Applied Algebra, vol.209, issue.1, pp.189-200, 2007.
DOI : 10.1016/j.jpaa.2006.05.028

M. M. Deza and M. Laurent, Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics, First softcover printing of the 1997 original [MR1460488]. [41] A. Dolzmann and T. Sturm. Redlog: Computer algebra meets computer logic, 2010.

C. Durvye, Algorithms for primary decomposition of zero-dimensional polynomial ideals given by an evaluation structure, 2008.
URL : https://hal.archives-ouvertes.fr/tel-00275219

C. Durvye and G. Lecerf, A concise proof of the Kronecker polynomial system solver from scratch, Expositiones Mathematicae, vol.26, issue.2, pp.101-139, 2008.
DOI : 10.1016/j.exmath.2007.07.001

URL : https://hal.archives-ouvertes.fr/hal-00682083

D. Eisenbud, Commutative algebra with a view toward algebraic geometry, 1995.

H. Everett, D. Lazard, S. Lazard, and M. , The Voronoi Diagram of Three Lines, Discrete & Computational Geometry, vol.1, issue.4, pp.94-130, 2009.
DOI : 10.1007/s00454-009-9173-3

URL : https://hal.archives-ouvertes.fr/inria-00186085

J. Faugère, A new efficient algorithm for computing Gr??bner bases (F4), Journal of Pure and Applied Algebra, vol.139, issue.1-3, pp.61-88, 1998.
DOI : 10.1016/S0022-4049(99)00005-5

J. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5 ), Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pp.75-83, 2002.

P. Festa, P. M. Pardalos, M. G. Resende, and C. C. Ribeiro, Randomized heuristics for the Max-Cut problem, Optimization Methods and Software, vol.25, issue.6, pp.1033-1058, 2002.
DOI : 10.1080/1055678021000090033

C. J. Friedrich, Alfred Weber's theory of location of industries, 1929.

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

A. Greuet and M. Safey-el-din, Deciding reachability of the infimum of a multivariate polynomial, Proceedings of the 36th international symposium on Symbolic and algebraic computation, ISSAC '11, pp.131-138, 2011.
DOI : 10.1145/1993886.1993910

URL : https://hal.archives-ouvertes.fr/hal-00744469

A. Greuet and M. Safey-el-din, Probabilistic and exact algorithm for the global optimization of a polynomial over a real algebraic set, 2013.

A. Greuet, F. Guo, M. Safey-el-din, and L. Zhi, Global optimization of polynomials restricted to a smooth variety using sums of squares, Journal of Symbolic Computation, vol.47, issue.5, pp.503-518, 2012.
DOI : 10.1016/j.jsc.2011.12.003

URL : https://hal.archives-ouvertes.fr/hal-00744605

F. Guo, M. Safey-el-din, and L. Zhi, Global optimization of polynomials using generalized critical values and sums of squares, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ISSAC '10, 2010.
DOI : 10.1145/1837934.1837960

URL : https://hal.archives-ouvertes.fr/hal-01292624

Q. Guo, M. Safey, E. 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, pp.197-204, 2013.
DOI : 10.1145/2465506.2465949

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

R. Hartley and A. Zisserman, Multiple view geometry in computer vision, 2000.
DOI : 10.1017/CBO9780511811685

J. Heintz and C. Schnorr, Testing polynomials which are easy to compute (Extended Abstract), Proceedings of the twelfth annual ACM symposium on Theory of computing , STOC '80, pp.262-272, 1980.
DOI : 10.1145/800141.804674

C. Helmberg and F. Oustry, Bundle Methods to Minimize the Maximum Eigenvalue Function, Handbook of Semidefinite Programming, pp.307-337, 2000.
DOI : 10.1007/978-1-4615-4381-7_11

C. Helmberg and F. , A Spectral Bundle Method for Semidefinite Programming, SIAM Journal on Optimization, vol.10, issue.3, pp.673-696, 2000.
DOI : 10.1137/S1052623497328987

D. Henrion and J. Lasserre, GloptiPoly, ACM Transactions on Mathematical Software, vol.29, issue.2, pp.165-194, 2003.
DOI : 10.1145/779359.779363

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

D. Henrion, M. ?ebek, and V. Ku?era, Positive polynomials and robust stabilization with fixed-order controllers, IEEE Transactions on Automatic Control, vol.48, issue.7, pp.1178-1186, 2003.
DOI : 10.1109/TAC.2003.814103

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

C. Hillar, Sums of squares over totally real fields are rational sums of squares, Proceedings of the American Mathematical Society, pp.921-930, 2009.
DOI : 10.1090/S0002-9939-08-09641-X

J. Hiriart-urruty and C. Lemarechal, Convex analysis and minimization algorithms ii, 1991.
DOI : 10.1007/978-3-662-06409-2

J. Hiriart-urruty and C. Lemaréchal, Convex analysis and minimization algorithms i, 1993.
DOI : 10.1007/978-3-662-02796-7

H. Hong, Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination, Papers from the international symposium on Symbolic and algebraic computation, ISSAC '92, pp.177-188, 1992.

H. Hong and M. Safey-el-din, Variant quantifier elimination, Journal of Symbolic Computation, vol.47, issue.7, pp.883-901, 2012.
DOI : 10.1016/j.jsc.2011.05.014

URL : https://hal.archives-ouvertes.fr/hal-00778365

T. Jacobi, A representation theorem for certain partially ordered commutative rings Testing sets for properness of polynomial mappings, Mathematische Zeitschrift Math. Ann, vol.23770, issue.3151, pp.259-2731, 1999.

Z. Jelonek, On the generalized critical values of a polynomial mapping, manuscripta mathematica, vol.110, issue.2, pp.145-157, 2003.
DOI : 10.1007/s00229-002-0320-x

Z. Jelonek and K. Kurdyka, On asymptotic critical values of a complex polynomial, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.2003, issue.565, pp.1-12, 2003.
DOI : 10.1515/crll.2003.101

G. Jeronimo, D. Perrucci, and E. P. Tsigaridas, On the Minimum of a Polynomial Function on a Basic Closed Semialgebraic Set and Applications, SIAM Journal on Optimization, vol.23, issue.1, pp.1-17, 2012.
DOI : 10.1137/110857751

URL : https://hal.archives-ouvertes.fr/hal-00776280

E. Kaltofen, On computing determinants of matrices without divisions, Papers from the international symposium on Symbolic and algebraic computation , ISSAC '92, pp.342-349, 1992.
DOI : 10.1145/143242.143350

E. L. Kaltofen, B. Li, Z. Yang, and L. Zhi, Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients, Journal of Symbolic Computation, vol.47, issue.1, pp.1-15, 2012.
DOI : 10.1016/j.jsc.2011.08.002

T. Krick, L. M. Pardo, and M. Sombra, Sharp estimates for the arithmetic nullstellensatz, Duke Mathematical Journal, vol.109, issue.3, pp.521-598, 2001.

J. Krivine, Anneaux pr??ordonn??s, Journal d'Analyse Math??matique, vol.12, issue.1, pp.307-326, 1964.
DOI : 10.1007/BF02807438

E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser Boston, 1984.
DOI : 10.1007/978-1-4612-5290-0

K. Kurdyka, P. Orro, and S. Simon, Semialgebraic Sard Theorem for Generalized Critical Values, Journal of Differential Geometry, vol.56, issue.1, pp.67-92, 2000.
DOI : 10.4310/jdg/1090347525

K. Kurdyka, P. Orro, and S. Simon, Semialgebraic Sard Theorem for Generalized Critical Values, Journal of Differential Geometry, vol.56, issue.1, pp.67-92, 2000.
DOI : 10.4310/jdg/1090347525

E. Landau, ???ber die Darstellung definiter Funktionen durch Quadrate, Mathematische Annalen, vol.62, issue.2, pp.272-285, 1906.
DOI : 10.1007/BF01449981

J. 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. Lasserre, A Sum of Squares Approximation of Nonnegative Polynomials, SIAM Review, vol.49, issue.4, pp.651-669, 2007.
DOI : 10.1137/070693709

M. Laurent, Sums of squares, moment matrices and optimization over polynomials. Emerging applications of algebraic geometry, pp.157-270, 2009.

M. Laurent, Sums of Squares, Moment Matrices and Optimization Over Polynomials, Emerging applications of algebraic geometry, pp.157-270, 2009.
DOI : 10.1007/978-0-387-09686-5_7

A. Lax and P. D. Lax, On sums of squares, Linear Algebra and its Applications, vol.20, issue.1, pp.71-75, 1978.
DOI : 10.1016/0024-3795(78)90031-9

D. Lazard and F. Rouillier, Solving parametric polynomial systems, Journal of Symbolic Computation, vol.42, issue.6, pp.636-667, 2007.
DOI : 10.1016/j.jsc.2007.01.007

URL : https://hal.archives-ouvertes.fr/hal-01148721

G. Lecerf, Une alternative aux méthodes de réécriture pour la résolution des systèmes algébriques, 2001.

G. Lecerf, Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers, Journal of Complexity, vol.19, issue.4, pp.564-596, 2003.
DOI : 10.1016/S0885-064X(03)00031-1

URL : https://hal.archives-ouvertes.fr/hal-00186727

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. Logar, A computational proof of the Noether normalization lemma, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp.259-273, 1989.
DOI : 10.1007/3-540-51083-4_65

S. Mccallum, An improved projection operation for cylindrical algebraic decomposition In Quantifier elimination and cylindrical algebraic decomposition (Linz, 1993), Texts Monogr, Symbol. Comput, pp.242-268, 1998.

D. Monniaux, On using sums-of-squares for exact computations without strict feasibility, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00487279

G. Moroz, Properness defects of projection and minimal discriminant variety, Journal of Symbolic Computation, vol.46, issue.10, pp.1139-1157, 2011.
DOI : 10.1016/j.jsc.2011.05.013

URL : https://hal.archives-ouvertes.fr/hal-01148309

T. S. Motzkin, The arithmetic-geometric inequality, Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, pp.205-224, 1965.

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

J. Nie, Polynomial optimization with real varieties. arXiv preprint arXiv:1211, 1940.

J. Nie, J. Nie, J. Demmel, and B. Sturmfels, An exact Jacobian SDP relaxation for polynomial optimization, Mathematical Programming, pp.225-255587, 2006.
DOI : 10.1007/s10107-011-0489-4

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

P. A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Dissertation (Ph.D.), California Institute of Technology, 2000.

P. A. Parrilo and B. Sturmfels, Minimizing polynomial functions Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol.60, pp.83-99, 2003.

H. Peyrl and P. A. Parrilo, Computing sum of squares decompositions with rational coefficients, Symbolic- Numerical Computations, pp.269-281, 2008.
DOI : 10.1016/j.tcs.2008.09.025

Y. Pourchet, Sur la représentation en somme de carrés des polynômes à une indéterminée sur un corps de nombres algébriques, Acta Arithmetica, vol.19, issue.1, pp.89-104, 1971.

V. Powers, Rational certificates of positivity on compact semialgebraic sets, Pacific Journal of Mathematics, vol.251, issue.2, pp.385-391, 2011.
DOI : 10.2140/pjm.2011.251.385

V. Powers and T. Wörmann, An algorithm for sums of squares of real polynomials, Journal of Pure and Applied Algebra, vol.127, issue.1, pp.99-104, 1998.
DOI : 10.1016/S0022-4049(97)83827-3

S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo, SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2004.

A. Prestel and M. Bradley, Representation of a real polynomial f(X) as a sum of 2m-th powers of rational functions, Ordered Algebraic Structures, pp.197-207, 1989.
DOI : 10.1007/978-94-009-2472-7_16

R. Quarez, Bounding the rational sums of squares over totally real fields. arXiv preprint, 2009.

B. Reznick, Some concrete aspects of Hilbert???s 17th Problem, Real algebraic geometry and ordered structures, pp.251-272, 1996.
DOI : 10.1090/conm/253/03936

R. M. Robinson, Some definite polynomials which are not sums of squares of real polynomials In Selected questions of algebra and logic (collection dedicated to the memory of A. I. Mal ? cev) (Russian), Sibirsk. Otdel, pp.264-282, 1973.

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

F. Rouillier, M. Roy, and M. , Finding at Least One Point in Each Connected Component of a Real Algebraic Set Defined by a Single Equation, Journal of Complexity, vol.16, issue.4, pp.716-750, 2000.
DOI : 10.1006/jcom.2000.0563

URL : https://hal.archives-ouvertes.fr/inria-00107845

M. Safey-el-din, Testing Sign Conditions on a Multivariate Polynomial and Applications, Mathematics in Computer Science, vol.1, issue.1, pp.177-207, 2007.
DOI : 10.1007/s11786-007-0003-9

URL : https://hal.archives-ouvertes.fr/inria-00105835

M. Safey-el-din, Computing the global optimum of a multivariate polynomial over the reals, ISSAC, pp.71-78, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01305635

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 at least one point in each connected component of a real algebraic set, Discrete Comput. Geom, vol.32, issue.3, pp.417-430, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00099962

M. Safey-el-din and E. Schost, A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets. arXiv preprint arXiv, pp.1307-7836, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00849057

M. Safey-el-din and L. Zhi, Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions, SIAM Journal on Optimization, vol.20, issue.6, pp.2876-2889, 2010.
DOI : 10.1137/090772459

URL : https://hal.archives-ouvertes.fr/inria-00419983

C. Scheiderer, Descending the ground field in sums of squares representations. arXiv preprint, 2012.

J. Schmid, On the degree complexity of Hilbert's 17th problem and the real nullstellensatz, Habilitationsschrift, 1998.

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

É. Schost, Computing parametric geometric resolutions Applicable Algebra in Engineering, Communication and Computing, vol.13, issue.5, pp.349-393, 2003.

H. Schramm and J. Zowe, A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, SIAM Journal on Optimization, vol.2, issue.1, pp.121-152, 1992.
DOI : 10.1137/0802008

M. Schweighofer, Algorithmische beweise für nichtnegativ-und positivstellensätze. Master's thesis, 1999.

M. Schweighofer, Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares, SIAM Journal on Optimization, vol.17, issue.3, pp.920-942, 2006.
DOI : 10.1137/050647098

A. Seidl and T. Sturm, A generic projection operator for partial cylindrical algebraic decomposition, Proceedings of the 2003 international symposium on Symbolic and algebraic computation , ISSAC '03, pp.240-247, 2003.
DOI : 10.1145/860854.860903

F. Severi, Sulle intersezioni delle varieta algebriche e sopra i loro caratteri e singolarita proiettive, Mem. Accad. Sci. Torino, vol.52, issue.2, pp.61-118, 1902.

F. Severi, La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica, Commentarii Mathematici Helvetici, vol.4, issue.1, pp.268-326, 1932.
DOI : 10.1007/BF01202721

I. Shafarevich, Basic Algebraic Geometry 1, 1977.

N. Z. Shor, An approach to obtaining global extrema in polynomial problems of mathematical programming, Kibernetika (Kiev), vol.136, pp.102-106, 1987.

G. Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry, Mathematische Annalen, vol.8, issue.2, pp.87-97, 1974.
DOI : 10.1007/BF01362149

A. W. Strzebo?ski, Cylindrical Algebraic Decomposition using validated numerics, Journal of Symbolic Computation, vol.41, issue.9, pp.1021-1038, 2006.
DOI : 10.1016/j.jsc.2006.06.004

A. Tarski, A Decision Method for Elementary Algebra and Geometry, 1998.
DOI : 10.1007/978-3-7091-9459-1_3

J. A. Todd, The geometrical invariants of algebraic loci, Congresso Internationale dei Matematici, p.93, 1928.

J. A. Todd, The Arithmetical Invariants of Algebraic Loci, Proceedings of the London Mathematical Society, pp.190-225, 1938.
DOI : 10.1112/plms/s2-43.3.190

H. V. Hà and T. S. Pha, Global optimization of polynomials using the truncated tangency variety and sums of squares, SIAM Journal on Optimization, vol.19, issue.2, pp.941-951, 2008.

H. Waki, S. Kim, M. Kojima, M. Muramatsu, and H. Sugimoto, Algorithm 883, ACM Transactions on Mathematical Software, vol.35, issue.2, 2009.
DOI : 10.1145/1377612.1377619