The Complex Zeros of Random Polynomials

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 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

Exact statistics of complex zeros for Gaussian random polynomials with real coefficients

k−point correlations of complex zeros for Gaussian ensembles of Random Polynomials of order N with Real Coefficients (GRPRC) are calculated exactly, following an approach of Hannay [5] for the case of Gaussian Random Polynomials with Complex Coefficients (GRPCC). It is shown that in the thermodynamic limit N → ∞ of Gaussian random holomorphic functions all the statistics converge to the their G...

Saddle points in the chaotic analytic function and Ginibre characteristic polynomial

Comparison is made between the distribution of saddle points in the chaotic analytic function and in the characteristic polynomials of the Ginibre ensemble. Realising the logarithmic derivative of these infinite polynomials as the electric field of a distribution of coulombic charges at the zeros, a simple mean-field electrostatic argument shows that the density of saddles minus zeros falls off...

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 → ∞.