Transcript Presentation 1 - Full

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.
Where
Is 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
()
n
i
,
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 1, 2, 3,..., n are the eigenvalues of A. Then,
det(A)  123 ...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
Topics to further research:
Real zeros of random matrices.
Random Polynomials and Riemann Sums.
Artwork gratefully borrowed from xkcd.com