Bounds on the Number of Real Solutions to Polynomial Equations

We use Gale duality for complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound $$\frac{e^{4}+3}{4}2^{\binom{k}{2}}n^{k}$$for the number of nonzero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of ℤⁿ of odd index. This bound only exceeds the bound for positive solutions by the constant factor (e⁴+3)/(e²+3) and it is asymptotically sharp for k fixed and n large.