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Anastasiya Tyurina

Constructing polynomial system with many positive solutions using tropical geometry (by Boulos El  Hilany)

The number of positive real solutions of a system of two polynomial equations in two unknowns with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. Using this method the author constructs  a system that has at most 6 positive solutions. He also shows that this bound is sharp. Moreover, using non-transversal intersections of tropical curves, he constructs a system that has  7 positive solutions.