Equilibrium Distribution of Zeros of Random Polynomials

Convergence of Random Zeros on Complex Manifolds

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N , orthonormalized on a regular compact set K ⊂ Cm, almost surely converge to the equilibrium measure on K as N → ∞.

Domain of attraction of normal law and zeros of random polynomials

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

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

Asymptotic zero distribution of random polynomials spanned by general bases

Zeros of Kac polynomials spanned by monomials with i.i.d. random coefficients are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal...

Equidistribution of zeros of random polynomials

We study the asymptotic distribution of zeros for the random polynomials Pn(z) = ∑n k=0 AkBk(z), where {Ak}k=0 are non-trivial i.i.d. complex random variables. Polynomials {Bk}k=0 are deterministic, and are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G bounded by an analytic curve. We show that the zero counting measures of Pn conv...