M. Aguiar, N. Bergeron, and F. Sottile, Combinatorial Hopf algebras and generalized Dehn???Sommerville relations, Compositio Mathematica, vol.142, issue.01, pp.1-30, 2006.
DOI : 10.1112/S0010437X0500165X

S. Armstrong and C. Hillar, Solvability of symmetric word equations in positive definite letters, Journal of the London Mathematical Society, vol.76, issue.3, pp.777-796, 2007.
DOI : 10.1112/jlms/jdm070

G. Baumslag, Some aspects of groups with unique roots, Acta Mathematica, vol.104, issue.3-4, pp.277-303, 1960.
DOI : 10.1007/BF02546390

G. Baumslag, On the residual nilpotence of certain one-relator groups, Communications on Pure and Applied Mathematics, vol.71, issue.5, pp.491-506, 1968.
DOI : 10.1002/cpa.3160210504

D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, Journal of Mathematical Physics, vol.16, issue.11, pp.2318-2325, 1975.
DOI : 10.1063/1.522463

G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc, pp.433-454, 1935.

B. Collins, K. Dykema, and F. Torres-ayala, Sum-of-Squares Results for Polynomials Related to??the??Bessis???Moussa???Villani Conjecture, Journal of Statistical Physics, vol.115, issue.5, 2009.
DOI : 10.1007/s10955-010-9959-y

S. Gao, Factoring multivariate polynomials via partial differential equations, Mathematics of Computation, vol.72, issue.242, pp.801-822, 2003.
DOI : 10.1090/S0025-5718-02-01428-X

D. Hagele, Proof of the Cases p??? 7 of the Lieb-Seiringer Formulation of the Bessis-Moussa-Villani Conjecture, Journal of Statistical Physics, vol.118, issue.6, pp.1167-1171, 2007.
DOI : 10.1007/s10955-007-9327-8

F. Hansen, Trace functions as Laplace transforms, Journal of Mathematical Physics, vol.47, issue.4, p.43504, 2006.
DOI : 10.1063/1.2186925

C. Hillar, Advances on the Bessis???Moussa???Villani trace conjecture, Linear Algebra and its Applications, vol.426, issue.1, pp.130-142, 2007.
DOI : 10.1016/j.laa.2007.04.005

C. Hillar and C. R. Johnson, Positive eigenvalues of generalized words in two Hermitian positive definite matrices, Novel Approaches to Hard Discrete Optimization of Fields Institute Communications, pp.111-122, 2003.
DOI : 10.1090/fic/037/07

C. Hillar and C. R. Johnson, Symmetric word equations in two positive definite letters, Proc. Amer, pp.945-953, 2004.

C. R. Johnson and C. Hillar, Eigenvalues of Words in Two Positive Definite Letters, SIAM Journal on Matrix Analysis and Applications, vol.23, issue.4, pp.916-928, 2002.
DOI : 10.1137/S0895479801387073

O. Kharlampovich and A. Myasnikov, Equations in a free Q-group, Transactions of the American Mathematical Society, vol.350, issue.03, pp.947-974, 1998.
DOI : 10.1090/S0002-9947-98-01798-X

O. Kharlampovich and A. Myasnikov, Elementary theory of free non-abelian groups, Journal of Algebra, vol.302, issue.2, pp.451-552, 2006.
DOI : 10.1016/j.jalgebra.2006.03.033

I. Klep and M. Schweighofer, Connes' embedding conjecture and sums of hermitian squares, Advances in Mathematics, vol.217, issue.4, pp.1816-1837, 2008.
DOI : 10.1016/j.aim.2007.09.016
URL : https://hal.archives-ouvertes.fr/hal-00372254

I. Klep and M. Schweighofer, Sums of Hermitian Squares and the BMV Conjecture, Journal of Statistical Physics, vol.10, issue.1???4, pp.739-760, 2008.
DOI : 10.1007/s10955-008-9632-x
URL : https://hal.archives-ouvertes.fr/hal-00310952

P. Landweber and E. Speer, On D. H??gele???s approach to the Bessis???Moussa???Villani conjecture, Linear Algebra and its Applications, vol.431, issue.8, pp.1317-1324, 2009.
DOI : 10.1016/j.laa.2009.05.002

J. Lawson and Y. Lim, Solving symmetric matrix word equations via symmetric space machinery, Linear Algebra and its Applications, vol.414, issue.2-3, pp.560-569, 2006.
DOI : 10.1016/j.laa.2005.10.035

K. J. Le-couteur, Representation of the function Tr(exp(A - ??B)) as a Laplace transform with positive weight and some matrix inequalities, Journal of Physics A: Mathematical and General, vol.13, issue.10, pp.3147-3159, 1980.
DOI : 10.1088/0305-4470/13/10/012

J. Ledlie, Representations of Free Metabelian ??? ?? -Groups, Transactions of the American Mathematical Society, vol.153, pp.307-346, 1971.
DOI : 10.2307/1995561

E. H. Lieb and R. Seiringer, Equivalent Forms of the Bessis???Moussa???Villani Conjecture, Journal of Statistical Physics, vol.115, issue.1/2, pp.185-190, 2004.
DOI : 10.1023/B:JOSS.0000019811.15510.27

W. Magnus, On a Theorem for Marshall Hall, The Annals of Mathematics, vol.40, issue.4, pp.764-768, 1939.
DOI : 10.2307/1968892

D. Marker, Model theory: an introduction, volume 217 of Graduate Texts in Mathematics, 2010.

P. Moussa, ) AS A LAPLACE TRANSFORM, Reviews in Mathematical Physics, vol.12, issue.04, pp.621-655, 2000.
DOI : 10.1142/S0129055X00000204

Z. Sela, DIOPHANTINE GEOMETRY OVER GROUPS VI: THE ELEMENTARY THEORY OF A FREE GROUP, GAFA Geometric And Functional Analysis, vol.16, issue.3, pp.707-730, 2006.
DOI : 10.1007/s00039-006-0565-8