A probabilistic Approach to polynomial Inequalities

Abstract

We define a probability measure on the space of polynomials over Rn 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 n 1 tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.