(We exclude polynomials with constant term 0, as their zeros, other than 0, are those of polynomials of lower degree with coefficients 0,1.) DefineFor each degree d, there are polynomials of degree d, and so W is a countable set.
There are few published results about W. In  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 . 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 .)
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 curvesand The curve is mapped to by . This mapping takes W to itself, since if , and z is a root of and , then is a root of . We show that all are enclosed strictly between and . >From this it follows that for all , where is the ``golden ratio.'' (The bound has been proved independently in different contexts by Flatto, Lagarias, and Poonen  and by Solomyak .) We also show that the line segment . However, and . Further, there is a constant such that if , , then . Thus the ``spike'' along the negative real axis that is visible in Fig. 1, connecting curves and with the exception of a small gap at -1, is due to zeros.
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 . For statements and references to general results, see .
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 ). 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 isand on average the zeros of tend to cluster near the zeros of .
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 .
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 setis 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  proves that the closure of the set of zeros of polynomials with coefficients 0, is connected. The thesis  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  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  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 .