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Bounds and locations for zeros


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



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