On the probability that the maximum of a polynomial is at an endpoint of an interval

In this article we study the probability that the maximum over a symmetric interval [−a, a] of a univariate polynomial of degree at most n is attained at an endpoint. We give explicit formulas for the degree n = 1, 2,3 cases. The formula for the degree 3 case leads to a lower bound for the true probability. Numerical experiments indicate that this lower bound is actually quite a good approximation. We finish with some numerical examples indicating that these ideas may be of use in the problem of finding average Markov factors for the bounds of the uniform norm of the derivative a polynomial in terms of the norm of the polynomial itself.