Bounds on the number of real solutions to polynomial equations

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