We show that there are fewer than
(e2+3)/4 2k(k-1)/2nk
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)(n+k-1)/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 Rn defined by a polynomial with n+k+1
monomials has at most
C(k)nk-1 compact connected components.
Our results hold for polynomials with real exponents.