On the number of minima of a random polynomial

Abstract

We give the upper bound 2 ( d - 1 ) ( n + 1 ) / 2 for the expected number of critical points of a normal random polynomial with degree at most d and n variables. Using the large deviation principle for the spectral value of large random matrices we obtain the bound K exp - n 2 ln 3 4 + n + 1 2 ln ( d - 1 ) for the expected number of minima of such a polynomial (here K is a positive constant). This proves that most normal random polynomials of fixed degree have only saddle points. Finally, we give a closed form expression for the expected number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions.