Contents
** Next:** A neighborhood of
**Up:** No Title
** Previous:** Introduction

A polynomial can have multiple zeros.
If is a **d**-th root of unity,
then is a zero of
and therefore a zero of for any **k** such that **d | k-1**.
Hence it is a zero of multiplicity 2 for , a polynomial in **P**.
Higher multiplicities can be obtained by iterating this procedure.
On the other hand, we do not know whether any that is not a root of unity can be a multiple root of any .
There do exist power series with coefficients **0,1** that have double
zeros **z** with **|z| <1**,
as will be shown in Section 3.
Inside a disk for **r < 1**, any polynomial
can have only a bounded number of zeros.
We prove a slightly more general result that will be used later
on.
### Proposition 2.1

Suppose that is a power series of the form
Then for any **r**, **0 < r < 1**, has
zeros in .

We next consider bounds on the size of .
Since for , it suffices to
consider .

### Theorem 2.1

Suppose that **z** satisfies **|z| < 1** and that for some power series
of the form .
Then
and , with equality if and only if
and .
Furthermore, there exists a such that if , then **z** is a negative real number.

The argument presented above is inefficient, and shows only that some value of is allowable.
With a little more care one could show by an extension of the
method used above that is allowable,
so that any with is real.
In Section 6 we present a variation of this method that uses
machine computation instead of careful estimates to establish rigorously that
is allowable.
Numerical evidence suggests that the minimal value of **|z|** over is about 0.734.
The method of Section 6 can be used to obtain estimates for the
minimal value of **|z|** over
that are as accurate as one desires.

By Proposition 3.1 of the next section,
.
Since is stable under and closed,
it follows that
.

In [8] it was shown that implies .
Theorem 2.1 immediately leads to the bound
for .
Numerical evidence suggests that for .
There are with .
The methods outlined in Section 6 can be used to obtain
precise bounds.

We can analyze inequality for **z** close to 1.
We find that for **z =1-x +iy** with **x** and **y** small,
**x > 0**, if then
fails,
so .
We next show that there are points in **W** which approach 1 along trajectories tangent to the real axis.

### Proposition 2.2

There exists a sequence of points such that as and

Contents
** Next:** A neighborhood of
**Up:** No Title
** Previous:** Introduction