  We show that there are fewer than (e^2+3) 2^(k choose 2) n^k/4 non-degenerate
positive solutions to a fewnomial system consisting of n polynomials in n
variables having a total of n+k+1 distinct monomials. This is significantly
smaller than Khovanskii's fewnomial bound of 2^(n+k choose 2)(n+1)^(n+k).
  We reduce the original system to a system of k equations in k variables which
depends upon the vector configuration Gale dual to the exponents of the
monomials in the original system. We then bound the number of solutions to this
Gale system. We adapt these methods to show that a hypersurface in the positive
orthant of R^n defined by a polynomial with n+k+1 monomials has at most
C(k)n^(k-1) compact connected components. Our results hold for polynomials with
real exponents.
