How many zeros has a random polynomial?

Goal:

The number (and the distribution) of the zeros of different ensembles of random polynomials have attracted the attention of physicists and mathematicians for many years.
Besides their implicit interest, they have a vast range of applications: algorithm complexity, quantum physics, etc.
The first class of studied random polynomials were the usual ones
pK(t)=a0 +a1x1+...+aKxK.
In this case it is known that, as K tends to infinity, the average and the variance of the number of zeros on the whole real line are asymptotic to 2/π log(K) and 4/π(1 - 2/π)log(K) respectively, for a large range of choices of the random coefficients an (centered and with variances equal to one). Nevertheless, with a convenient choice of the coefficients (centered but with different variances of a very concrete form) it is known that the asymptotic number of zeros of such a polynomial is √K; that is really much more than the preceeding value.
Besides, there are results that state that in case of systems of complex homogeneized polynomials, the roots are surprisingly isotropically distributed on the sphere.
Other classes of studied random polynomial include the trigonometric polynomials and the hypergeometric polynomials.
For trigonometric polynomials, Central Limits Theorems for their number of zeros have been established recently in the situation of independent standard Gaussian coefficients. In the case of the hyperbolic polynomials there are known asymptotics for the mean number of zeros under different choices of the random coefficients (all Gaussian for the moment).

The aim of this mémoire is (at least) threefold:
- to review the existing bibliography on the subject (with a special emphasis on trigono- metric polynomials).
- to study the techniques that have been used so far.
- to make some numerical simulations in order to `confirm' some of the already known results and to `discover' some new ones.

PDF description of the project

Supervisor: Federico Dalmao.

Difficulty level: Master student project

Results: See the report.

FSCT -- University of Luxembourg