A probabilistic approach to polynomial inequalities

We define a probability measure on the space of polynomials over ? ⁿ in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities.Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of ? ₁ ⁿ tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.