We use Gale duality for complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound  

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.

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