Some computations of expected number of real zeros of random polynomials

Expected Number of Local Maxima of Some Gaussian Random Polnomials

Expected Number of Real Zeros for Random Linear Combinations of Orthogonal Polynomials

We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only (2/π + o(1)) logn expected real zeros in terms of the degree n. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations...

Expected Discrepancy for Zeros of Random Polynomials

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like √ logn/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of t...

Expected Number of Local Maxima of Some Gaussian Random Polynomials

Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj − Aj−1, j = 0, 1, 2, · · · are independent, A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of lo...

expected number of local maxima of some gaussian random polnomials