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Zeros of polynomials with random coefficients occur in many scientific and engineering problems. A general overview of the subject and references can be found in the book of Bharucha-Reid and Sambandham [4], which is the basic reference on this topic. There is a wealth of information about distribution of zeros in the complex plane and on the real line. Almost all of the results are for coefficients chosen independently from a common distribution that is continuous, and usually Gaussian. In this paper we consider zeros of polynomials with

(We exclude polynomials with constant term 0,
as their zeros, other than 0, are those of polynomials of lower degree with coefficients **0,1**.)
Define

There are few published results about **W**.
In [8] it was shown that for all . This was used to prove that if is a prime
for some , then is irreducible over the
rationals.
(For further results relating zeros to irreducibility, see
[12].
It is conjectured that almost all are
irreducible, but this is still open.
This is in contrast to the case of fixed degree polynomials when
the range over which the coefficients are allowed to run
increases.
There it is known that almost all polynomials are not only
irreducible, but also have as their Galois group.
For latest results and references on this topic, see
[14].)

Our results are best illustrated by pictures of zeros.
Figure 1 shows all zeros of
the
polynomials with
coefficients **0,1** of degrees 12 and less,
and with constant term 1,
except for the negative real zeros that lie are .
We show that **W** lies
between the curves

Since polynomials in **P** have nonnegative coefficients,
.
However, since for every root of unity , , where denotes the
closure of **W**.
We answer a question posed by J. H. Conway and Richard Parker
about the behavior of **W** near 1 by proving there exist points
such that , so that these
points come in tangent to the **x**-axis.

Figure 2 shows the zeros of polynomials of degree 12
that are close to **z=1**.
Figure 3
shows zeros of polynomials of all degrees
that fall in a certain small region of the complex
plane.
Figures 4,
5, and
6
show pictures of parts of .
The region depicted in
Figure 4is the same as that of
Figure 3.
Section 6 explains how these pictures were created.

Theorem 2.1 of
Section 2, which says that **W** is
contained between and , is not best possible.
The only points of that are in are
1, , .
In
Section 6 we will show how to obtain more precise bounds for
**W**.
However, because of the fractal nature of **W**, there is no simple
description of its shape.

Many features visible in the graphs can be explained (at least
heuristically, and often rigorously) by using known results or
methods.
When one
graphs zeros of any single polynomial with coefficients
**0** and **1**, most of them are close to the unit circle
**|z|=1**
and they are equidistributed in angles, so that the first
quadrant, for example, has close to of the total.
This phenomenon is true for all polynomials whose coefficients
do not vary much, as follows from results originating with
Erdös and Turán [11].
For statements and references to general results, see
[4].

The expected number of real roots of a random polynomial (which have to be negative for ) grows logarithmically with **n**,
as was first noted by Kac and Rice (see [4]).
Furthermore, the variance is small.

In
Figures 1 and 2, there is a perceptible clustering of zeros.
This is a reflection of the
``averaging phenomenon'' for roots
of random polynomials
[4,15], and again is not special to
**0,1** coefficients.
The ``average'' of the polynomials of degree **n** that are in **P**
is

Figures 1 and 2 show several large ``holes,''
which contain either just one or no zeros.
These holes are usually centered at algebraic integers of low degree and small height (i.e.,
algebraic integers that satisfy
polynomial equations with small integral coefficients).
The most prominent of the holes are at the roots of unity,
such as **-1** and **i**.
As one computes zeros of polynomials of increasing degrees, the
large holes in Figures 1 and 2 fill up.
However, there are other holes, such as those visible in
Figures
3,
4,
5, and
6
that are free of zeros even when the degree increases.

We show in Section 3 that there is an open neighborhood of that is in . In Section 4 we prove that is connected. The more involved argument in Section 5 proves that is path connected. Since the unit circle is contained in , but , is clearly not simply connected. Numerical experiments suggest that has ``holes'' in it besides the big hole containing 0. (That is, has more than 2 connected components.) In particular, the disk of radius centered at appears to be part of such a hole. This hole and some neighboring ones are pictured in Figures 5, and 6. Other, even larger holes, can be seen in Figures 3 and 4.

**W** has a fractal appearance that is reminiscent of some of the
Julia sets
[1,10].
In
Section 6
we sketch arguments that explain how this arises.
However, we do not have estimates for such interesting parameters
as the Hausdorff dimension of the boundary of .

In contrast to our result that is path connected, the Mandelbrot set is only known to be connected, although it is conjectured to be path connected [1,10]. Our methods are simpler than those used to study the connectedness of the Mandelbrot set. They are similar to the techniques developed for investigating iterated function systems [1].

Results
similar to those for polynomials with
**0,1** coefficients can also be obtained for other families of
polynomials with a small set of possible coefficients.
For example, for
coefficients,
pictures of zeros are qualitatively similar to those of
**0,1** polynomials.
There is symmetry about the imaginary axis as well as the
real axis (corresponding to changing the variable **z** to **-z**).
There are two ``spikes''
of zeros along the real axis that fill the intervals
and , while there are no other
zeros in or for some .
For polynomials with cubic roots of unity as coefficients, there are no
``spikes,''
but the zeros still have a fractal appearance.

The set

is the set of zeros of power series Since for all , it is sufficient to study , , and in some ways it is more natural to deal with the above power series.Some of our methods and results are similar to those of Thierry Bousch [5,6], whose work was brought to our attention by D. Zagier. The report [5] proves that the closure of the set of zeros of polynomials with coefficients 0, is connected. The thesis [6] contains, along with a variety of other results, general methods for studying similar problems. In the area where our work overlaps [5,6], we obtain a somewhat stronger result by proving path connectivity.

Boris Solomyak [16] has studied zeros of power series of the form , but with the , , allowed to take any real values in the interval . He shows that the bound holds there as well, and that there is a ``spike'' of real zeros along the negative real axis. However, the zeros of Solomyak's functions are substantially different from those we investigate. For example, he shows that segments of the boundary he investigates have everywhere dense sets of points where a tangent exists, as well as everywhere dense sets of points with no tangent. There are also no holes in Solomyak's set of zeros.

The paper of Brenti, Royle, and Wagner [7] discusses various properties of chromatic polynomials. While it is not directly related to our work, the numerical evidence it presents shows that zeros of chromatic polynomials may also exhibit fractal behavior. This may also be true for the partition function zeros of [3].

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