Systems of rational polynomial equations have polynomial size approximate zeros on the average

A new technique for the Geometry of Numbers is exhibited. This technique provides sharp estimates on the number of bounded height rational rational points in subsets of projective space whose “projective cone” is semi–algebraic. This technique improves existing techniques as the one introduced by H. Davenport in [15]. As main outcome, we conclude that systems of rational polynomial equations of bounded bit length have polynomial size approximate zeros on the average. We also conclude that the average number of projective real solutions of systems of rational polynomial equations of bounded bit length equals the square root of the Bézout Number of the given system.