D. N. Bernstein, The number of roots of a system of equations, Funkts. Anal. Prilozh, vol.9, issue.3, pp.1-4, 1975.

F. Bihan and I. Soprunov, Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes, Adv. Geom, vol.19, issue.4, pp.527-540, 2019.

R. Bieri and J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math, vol.347, pp.168-195, 1984.

M. A. Cueto, Implicitization of Surfaces via Geometric Tropicalization

C. and M. Sombra, The Newton polygon of a rational plane curve, Mathematics in Computer Science, vol.4, pp.3-24, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00359281

J. Draisma, A tropical approach to secant dimensions, J. Pure Appl. Algebra, vol.212, issue.2, pp.349-363, 2008.

M. Einsiedler, M. M. Kapranov, and D. Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math, vol.601, pp.139-157, 2006.

I. Z. Emiris and I. S. Kotsireas, Implicit polynomial support optimized for sparseness, Proc. Intern. Conf. Computational science appl.: Part III, pp.397-406, 2003.

A. Esterov and A. Khovanskii, Elimination theory and Newton polytopes, Funct. Anal. Other Math, vol.2, issue.1, pp.45-71, 2008.

J. P. Jouanolou, Théorèmes de Bertini et applications, Progress in Math, vol.42, 1983.

P. Hacking, S. Keel, and J. Tevelev, Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math, vol.178, issue.1, pp.173-227, 2009.

M. I. Herrero, G. Jeronimo, and J. Sabia, On degree bounds for the sparse Nullstellensatz, J. Algebra, vol.534, pp.531-544, 2019.

A. Jensen and J. Yu, Computing Tropical Resultants, J. Algebra, vol.387, pp.287-319, 2013.

F. V. Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, Progress in Mathematics, vol.181, pp.381-456, 2000.

T. Markwig, A field of generalized Puiseux series for tropical Geometry, Rend. Semin. Mat. Univ. Politec. Torino, vol.68, issue.1, pp.97-92, 2010.

B. Poonen, Maximally complete fields, Enseign. Math., II. Sér, vol.39, issue.1, pp.87-106, 1993.

B. Sturmfels, The Newton polytope of the resultant, J. Algebr. Combinatorics, vol.3, pp.207-236, 1994.

B. Sturmfels and J. Tevelev, Elimination theory for tropical varieties, Math. Res. Lett, vol.15, issue.3, pp.543-562, 2008.

B. Sturmfels, J. Tevelev, and J. Yu, The Newton polytope of the implicit equation, Mosc. Math. J, vol.7, issue.2, p.351, 2007.

B. Sturmfels and J. Yu, Minimal polynomials and sparse resultants, pp.317-324, 1992.

B. Sturmfels and J. Yu, Tropical implicitization and mixed fiber polytopes. Software for algebraic geometry, Math. Appl, vol.148, pp.111-131, 2008.

, E-mail address: alidick@dm.uba.ar E-mail address: iherrero@dm.uba.ar INRIA Sophia Antipolis Méditerranée, AROMATH, 2004 route des Lucioles, 1428.