On the number of minima of a random polynomial

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

On the average number of sharp crossings of certain Gaussian random polynomials

2 6 A pr 2 00 7 On the number of minima of a random polynomial ∗

We give an upper bound in O(d) for the 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 O “ exp(−βn + n 2 log(d− 1)) ” (β is a positive constant independent on n and d) for the number of minima of such a polynomial. This proves that most normal ra...

ar X iv : m at h / 07 02 36 0 v 1 [ m at h . N A ] 1 3 Fe b 20 07 On the number of minima of a random polynomial ∗

We give an upper bound in O(d) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d)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 hy...

Expected Number of Local Maxima of Some Gaussian Random Polnomials