On the number of minima of a random polynomial

We give an upper bound in O(d(n+1)/2 ) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d(n+1)/2 )Pn, where Pn is the (unknown) measure of the set of symmetric positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions.

• Type: Journal
• Year: 2008
• Where: JC
• Authors: Jean-Pierre Dedieu, Gregorio Malajovich
• Keywords: Gaussian Orthogonal Ensemble, JC 2008, Normal Random Polynomial, Symmetric Positive Matrices