Counting Real Connected Components of Trinomial Curve Intersections and   m-nomial Hypersurfaces
  We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial. 
Tien-Yien Li (Michigan State University), J. Maurice Rojas (Texas A&M   University), Xiaoshen Wang (University of Arkansas a...