Checking some Asymptotic Properties of Real and Complex Roots of Random Algebraic Polynomials

Henryk Zawadzki

University of Economics, Katowice, Poland

Abstract

When parameters of the algebraic polynomial

(Subscript[f, n])(z)=(∑_(j=0)^n)(Subscript[a, j])(z^j)

are random variables, the number (Subscript[ν, n]) of all real roots of (1) is also a random variable. M. Kac derived a formula for the expected value of (Subscript[ν, n]), assuming that the parameters (Subscript[a, j]) ( j = 0, ..., n) are i.i.d. standard normal. The formula is of the form

E((Subscript[ν, n]))=(∫_(-∞)^∞)(Subscript[g, n])(t)dt

with so called intensity functions

(Subscript[g, n])(t)=(1/π)(Sqrt[(1/((((t^2)-1))^2))-(((((n+1))^2) (t^(2n)))/((((t^(2n+2))-1))^2))])

It is also known that

E((Subscript[ν, n]))=(2/π)ln n +(2/(π n))+(Subscript[C, 1])+O((n^(-2)))

with (Subscript[C, 1]) ≈ 0.6257358072...

Using  functions  Manipulate and CountRoots we show how the shape of the graphs of functions (3) changes when n is increasing. Calculating the values of  E((Subscript[ν, n]))from (2) and the values of the sequence (Subscript[c, n])=(2/π) ln n +(2/(π n))+(Subscript[C, 1]) we also show that for increasing n we have   ((E((Subscript[ν, n])))/(Subscript[c, n]))->1 and |E((Subscript[ν, n]))-(Subscript[c, n])|->0. Generating a big number of random polynomials with i.i.d. standard normal coefficients, we "check" the Kac formula comparing the theoretical values of  E((Subscript[ν, n]))given by (2) with expected values of (Subscript[ν, n]) obtained from the simulation. With a help of user defined function  Countroots[poly_,{α_,β_}] and some standard functions offered by Mathematica we illustrate the fact, that all the roots of  (1) cluster near unit circle |z| = 1 and are (asymptotically) uniformly distributed in argument in the sense of convergence ((E[(Subscript[ν, n]))S(α, β))])/n)->((β-α)/(2π)), where (Subscript[ν, n])(S(α,β))denotes the number of roots of (1) lying in the sector S(α,β)={z: α ≤ Arg(z)<β} of the complex plane, that does not intersect the real axis.