On the number of minima of a random polynomial

Authors:
Jean-Pierre Dedieu;Gregorio Malajovich
Affiliations:
Institut de Mathématiques, Université Paul Sabatier, 31062 Toulouse Cedex 9, France;Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970 Rio de Janeiro, RJ, Brazil
Venue:
Journal of Complexity
Year:
2008

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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 boundKexp-n^2ln34+n+12ln(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.