Random Polynomials: crossings of levels, and turning points.
Since secondary school, everyone remembers quadratic and cubic equations. In other words, everybody has met an algebraic polynomial of degree 2 or 3 at least once in their life. Those who know some more mathematics are aware that there is no general formula for solving algebraic equations of degree higher than four. However, many questions related to these equations, as well as polynomials, of higher degrees can be approached and answered without solving the equation explicitly. For example, how would you like this one: "How many solutions in average do equations of a given degree have?"
Besides purely mathematical interest, results in this direction may be useful for physicists, engineers and even economists, since zeros of polynomials can represent lines of a physical spectrum, eigenvalues of important matrices, and so on.
The project considers several types of polynomials including algebraic and trigonometric ones. Randomness is introduced by putting polynomial coefficients to be random variables of known probability distributions. The work concentrates on establishing asymptotic regularities for the average number of zeros of a random polynomial of a high degree. Both real and complex zeros are dealt with. Related mathematical problems are tackled, e.g. the number of extrema or level crossings of a polynomial, etc.
Personnel Involved
First Supervisor: Farahmand, K Prof
Second Supervisor: Wang, H Dr
Collaboration: This project does not involve collaboration with another establishment
Synopsis:
One classical subject of mathematics is quadratic and cubic equations with deterministic coefficients. However, in many real life problems the coefficients are random variables, as they are when determined by an experiment. In this project we look at the behaviour of polynomials, usually with large degree with random coefficients. In particular we study the number of real zeros which give an indication of oscillation properties of polynomials.