Constructing polynomial systems with many positive solutions using tropical geometry

The number of positive solutions to a system of two polynomials in two variables
 defined over the field of real numbers with a total of five distinct monomials
 cannot exceed 15. All previously known examples have at most 5 positive solutions.
 In this talk, I am going to present a construction of a system as above having 7
 positive solutions. This is achieved using tools developed in tropical geometry.
 When the corresponding tropical hypersurfaces intersect transversally, one can
 easily estimate the positive solutions to the system using the classical combinatorial
 patchworking for complete intersections. This generalization is applied to construct a
 system as above having 6 positive solutions. It turns out that this bound is sharp.
 The main result is proved using non-transversal intersections of tropical curves.