  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 counted only non-degenerate roots and even then gave much
larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound
is sharp, allows real exponents, 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 new bounds on the number of real
connected components of fewnomial hypersurfaces.
