The probabilistic approach and the BKK Theorem

The BKK Theorem states that the number of complex roots of a complex polynomial system can be counted with the help of the volume of the Newton polytope. But what if we want to count real roots of real systems? There is no “generic” number of solution in that case. One can then try with a probabilistic approach and count the “average” number of solutions of “random” systems. In this talk, I will show, in the Gaussian case, how this leads to a result similar to the classical BKK theorem but where the volume of the Newton polytope is not the Euclidean volume but the Riemannian volume for a certain (explicit) metric. I will then explain the conscequences in terms of monotonicity of the number of solution and how this invalidates a local version of a conjecture by Bürgisser.