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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 
.
[Proof]
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.
[Proof]
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
[Proof]
Contents
Next: A neighborhood of
Up: No Title
Previous: Introduction