S. Basu, R. Pollack, and M. Roy, At the moment, we are not able to provide any SOS decomposition for all such benchmarks Our SDPbased algorithm relies on the high-precision solver SDPA-GMP [25] but it is still challenging to obtain precise values of eigenvalues/vectors of SDP output matrices. Another advantage of this technique is its ability to perform global polynomial optimization. A topic of interest would be to obtain the same feature with the two current algorithms. We also plan to develop extensions to the non-polynomial case On the combinatorial and algebraic complexity of quantifier elimination, Journal of the ACM (JACM), issue.6, pp.431002-1045, 1996.

S. Basu, R. Pollack, and M. Roy, Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics), 2006.
DOI : 10.1007/978-3-662-05355-3

A. L. Cauchy, Calcul des indices des fonctions, pp.176-229
DOI : 10.1017/CBO9780511702501.013

S. Chevillard, J. Harrison, M. Joldes, and C. Lauter, Efficient and accurate computation of upper bounds of approximation errors, Theoretical Computer Science, vol.412, issue.16, pp.1523-1543, 2011.
DOI : 10.1016/j.tcs.2010.11.052

URL : https://hal.archives-ouvertes.fr/ensl-00445343

G. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, pp.134-183, 1975.
DOI : 10.1007/3-540-07407-4_17

J. Gathen and J. Gerhard, Modern Computer Algebra, 1999.

A. Girard, Invention nouvelle en l'algébre, Blauew, p.1629

A. Greuet and M. Safey-el-din, 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

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, pp.197-204, 2013.
DOI : 10.1145/2465506.2465949

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

J. Harrison, HOL Light: A tutorial introduction, Lecture Notes in Computer Science, vol.1166, pp.265-269, 1996.
DOI : 10.1007/BFb0031814

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.42

D. Henrion and M. Naldi, Safey El Din. Spectra-a maple library for solving linear matrix inequalities in exact arithmetic, 2016.

N. J. Higham, Accuracy and Stability of Numerical Algorithms: Second Edition, 2002.
DOI : 10.1137/1.9780898718027

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

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

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

T. Lickteig and M. Roy, Sylvester?Habicht Sequences and Fast Cauchy Index Computation, Journal of Symbolic Computation, vol.31, issue.3, pp.315-341, 2001.
DOI : 10.1006/jsco.2000.0427

URL : http://doi.org/10.1006/jsco.2000.0427

V. Magron, X. Allamigeon, S. Gaubert, and B. Werner, Formal proofs for Nonlinear Optimization, Journal of Formalized Reasoning, vol.8, issue.1, pp.1-24, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00985675

K. Mehlhorn, M. Sagraloff, and P. Wang, From approximate factorization to root isolation with application to cylindrical algebraic decomposition, Journal of Symbolic Computation, vol.66, pp.34-69, 2015.
DOI : 10.1016/j.jsc.2014.02.001

URL : http://arxiv.org/abs/1301.4870

S. Melczer and B. Salvy, Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC '16, pp.333-340, 2016.
DOI : 10.1145/2930889.2930913

URL : https://hal.archives-ouvertes.fr/hal-01310691

M. Mignotte, Mathematics for Computer Algebra, 1992.
DOI : 10.1007/978-1-4613-9171-5

M. Nakata, A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD., 2010 IEEE International Symposium on Computer-Aided Control System Design, pp.29-34, 2010.
DOI : 10.1109/CACSD.2010.5612693

P. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, 2000.

H. Peyrl and P. Parrilo, Computing sum of squares decompositions with rational coefficients, Theoretical Computer Science, vol.409, issue.2, pp.269-281, 2008.
DOI : 10.1016/j.tcs.2008.09.025

URL : http://doi.org/10.1016/j.tcs.2008.09.025

Y. Pourchet, Sur la reprsentation en somme de carrs des polynmes une indtermine sur un corps de nombres algbriques, Acta Arithmetica, vol.19, issue.1, pp.89-104, 1971.

A. Prestel and C. Delzell, Positive Polynomials: From Hilberts 17th Problem to Real Algebra, 2001.
DOI : 10.1007/978-3-662-04648-7

A. Rantzer and P. A. Parrilo, On convexity in stabilization of nonlinear systems, Proceedings of the 39th IEEE Conference on Decision and Control, pp.2942-2945, 2000.

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

M. Schweighofer, Algorithmische Beweise für Nichtnegativ-und Positivstellensätze, 1999.

A. Strzebonski and E. Tsigaridas, Univariate real root isolation in an extension field, Proceedings of the 36th international symposium on Symbolic and algebraic computation, ISSAC '11, pp.321-328, 2011.
DOI : 10.1145/1993886.1993934

URL : https://hal.archives-ouvertes.fr/hal-01248390

W. Swokowski, Fundamentals of College Algebra, pp.216-221, 1989.

D. Y. Yun, On square-free decomposition algorithms, Proceedings of the third ACM symposium on Symbolic and algebraic computation , SYMSAC '76, pp.26-35, 1976.
DOI : 10.1145/800205.806320