New fewnomial upper bounds from Gale dual polynomial system

We show that there are fewer than $\frac{e^2+3}{4}2^{\binom{k}{2}}n^k$ 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 $2^{\binom{n+k}{2}}(n+1)^{n+k}$. 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 $\mathbb R^n$ 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.