Bounds on the number of real solutions to polynomial equations

    We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e⁴+3)/4 2^k(k-1)/2n^k for the number of non-zero real solutions to a system of n polynomials in n variables having n+k1 monomials whose exponent vectors generate a subgroup of Zⁿ of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e⁴+3)/(e²+3) and it is asymptotically sharp for k fixed and n large.