Probability of Real Zeros with Random

Polynomials

By Anne Calder & Alan Cordero

What is a Random Polynomial?

A random polynomial, is defined here to be a polynomial with independent standard normally distributed coefficients.

What does it mean to have a real zero?

If you have a polynomial of degree one:

It can always be solved for a real zero.

What about polynomials with degree greater than one?

Polynomials with a degree greater than one cannot always be solved for a real zero.

Taking a look at the quadratic formula:

 

We see that a polynomial of degree two can only be solved

for real zeros when:

Mark Kac, a polish mathematician, developed a formula for the expected number of zeros from polynomials.

WhereIs the expected number of real zeros.

The above formula is complicated. If we change how we choose our coefficients we can

get a much nicer formula….

If we change our coefficients to be not only normal with mean zero,

but with the variance of the ith coefficient equal to ,( )ni

we get a much nicer formula.

.

Expected Number of Real Zeros

Expected Proportion of Real Zeros

Zeros of Polynomials and Companion Matrices

Given a random polynomial:

You can build a Companion Matrix:Where the zeros of p(x) are the eigenvalues of A.

Eigenvalues:Suppose that are the eigenvalues of A. Then,

1,2,3,...,n

det(A)123...n

So…. Who Cares?

Among many other applications, random polynomials and their real roots have a part to play in stochastic processes.

The structure of random processes has been the subject of intensive investigation over the last few decades. One of the characteristics of the random process which has been greatly studied concerns the zeros and level crossing behavior, of random processes. The behavior is not only of profound physical and theoretical interest, but is also of considerable practical importance. They have found applications in many areas of physical science including hydrology, seismology, meteorology, reliability theory, aerodynamics and structural engineering.

Farahmand. Topics in random polynomials. USA: Addison Wesley Longman, Inc. 1998

Real zeros of random matrices.

Random Polynomials and Riemann Sums.

Topics to further research:

Artwork gratefully borrowed from xkcd.com