H. [. Anai and . Yanami, SyNRAC: A Maple-Package for Solving Real Algebraic Constraints, Part I, Lecture Notes in Comput. Sci, vol.2657, pp.828-837, 2003.
DOI : 10.1007/3-540-44860-8_86

M. [. Boyd and . Grant, Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pp.95-110, 2008.

M. [. Boyd, Y. Grant, and . Ye, Disciplined convex programming, Global optimization, Nonconvex Optim. Appl, vol.84, pp.155-210, 2006.

S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, A tutorial on geometric programming, Optimization and Engineering, vol.141, issue.1-2, pp.67-127, 2007.
DOI : 10.1007/b98874

]. G. Ble06 and . Blekherman, There are significantly more nonnegative polynomials than sums of squares, Israel J. Math, vol.153, pp.355-380, 2006.

J. Squares, G. Amer, P. A. Blekherman, R. R. Parrilo, S. Thomas et al., Semidefinite optimization and convex algebraic geometry Convex optimization On the computational complexity of membership problems for the completely positive cone and its dual Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube Geometric programming: Theory and application, SIAM and the Mathematical Optimization Society Amoebas, nonnegative polynomials and sums of squares supported on circuits, pp.617-635, 1967.

A. [. Fidalgo and . Kovacec, Positive semidefinite diagonal minus tail forms are sums of squares, Mathematische Zeitschrift, vol.283, issue.3-4, pp.3-4, 2011.
DOI : 10.1007/BF01442738

URL : http://estudogeral.sib.uc.pt/jspui/bitstream/10316/11199/1/Positive%20semidefinite%20diagonal%20minus%20tail%20forms%20are%20sums%20of%20squares.pdf

M. Ghasemi, J. B. Lasserre, and M. Marshall, Lower bounds on the global minimum of a polynomial, Computational Optimization and Applications, vol.8, issue.2, pp.387-402, 2014.
DOI : 10.1007/s11081-007-9001-7

M. [. Ghasemi and . Marshall, Lower Bounds for Polynomials Using Geometric Programming, SIAM Journal on Optimization, vol.22, issue.2, pp.460-473, 2012.
DOI : 10.1137/110836869

URL : http://math.usask.ca/%7Emarshall/lbgp17.pdf

H. D. Henrion, J. B. Lasserre, and J. Löfberg, basic closed semialgebraic set using geometric programming, 2013, Preprint, arxiv:1311.3726 GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw, vol.24, pp.4-5, 2009.

S. Iliman and T. De-wolff, Amoebas, nonnegative polynomials and sums of squares supported on circuits Lower bounds for polynomials with simplex newton polytopes based on geometric programming, Res. Math. Sci. SIAM J. Optim, vol.39, issue.26 2, pp.1128-1146, 2016.
DOI : 10.1137/140962425

URL : http://arxiv.org/pdf/1402.6185

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

]. J. Las01 and . Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim, vol.11, issue.3, pp.796-817, 2000.

M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry, Imperial College Press Optimization Series Math. Appl, vol.1, issue.149, pp.157-270, 2009.
DOI : 10.1007/978-0-387-09686-5_7

URL : http://homepages.cwi.nl/~monique/files/moment-ima.pdf

J. [. Nie, B. Demmel, and . Sturmfels, Minimizing Polynomials via Sum of Squares over the Gradient Ideal, Mathematical Programming, vol.13, issue.3, pp.587-606, 2006.
DOI : 10.4153/CMB-2003-054-7

URL : http://arxiv.org/pdf/math/0411342v2.pdf

]. J. Nie13a and . Nie, Certifying convergence of Lasserre's hierarchy via flat truncation, Math. Program, vol.142, issue.12, pp.485-510, 2013.

, [Nie14] , Optimality conditions and finite convergence of Lasserre's hierarchy Nesterov and A. Nemirovskii, Interior point polynomial algorithms in convex programming, Oxl11] J. Oxley, Matroid theory, pp.225-255, 1994.

]. A. Pav-`-pav-`-13, J. Papachristodoulou, G. Anderson, S. Valmorbida, P. Prajna et al., SOSTOOLS: Sum of squares optimization toolbox for MATLAB, http://arxiv.org/abs Available from http, sostools. [PP08] H. Peyrl and P.A. Parrilo, Computing sum of squares decompositions with rational coefficients, pp.269-281, 1310.

B. [. Parrilo and . Sturmfels, Minimizing polynomial functions, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc, vol.60, pp.83-99, 2001.
DOI : 10.1090/dimacs/060/08

]. B. Rez78 and . Reznick, Extremal PSD forms with few terms, Duke Math Some concrete aspects of Hilbert's 17th Problem, Real algebraic geometry and ordered structures (Baton Rouge Global optimization of polynomials using gradient tentacles and sums of squares, J. Contemp. Math. Amer. Math. Soc. SIAM J. Optim, vol.45, issue.253 3, pp.363-374, 1978.

. Wkk-`-wkk-`-09-]-h, S. Waki, M. Kim, M. Kojima, H. Muramatsu et al., Algorithm 883: sparsePOP? a sparse semidefinite programming relaxation of polynomial optimization problems, ACM Trans. Math. Software, vol.35, issue.2, pp.15-28, 2009.

M. Dressler and . Goethe-universität, Postfach 11 19 32, 60054 Frankfurt am Main, Germany E-mail address: dressler@math.uni-frankfurt.de Sadik Iliman Frankfurt am Main, Germany E-mail address: sadik.iliman@gmx.net Timo de Wolff