God Plays Dice

Here's a little thing I thought of a few days ago. Consider the following identities for the sums of powers:

(okay, that's kind of stupid, but you have to start somewhere...),


(the right-hand side might be more familiar as n(n+1)(2n+1)/6), and

(the right-hand side here is,coincidentally the square of (1+2+...+k). For each choice of exponent we get a different polynomial in the numerator. They all factor into linear terms... that doesn't keep up, though. For example,

Still, one wonders -- what are the roots of these polynomials? (The first thought is that they're always in the interval [-1, 0], but that's pretty quickly disproven by considering the sum of 5th powers.)

Some computation shows that the patterns of zeroes in the complex plane are both symmetric around the real axis (no surprise there; zeroes come in complex conjugate pairs!) and around the line y = -1/2 (a bit more surprising). So you think to plot them, and you get something that looks like this plot for the polynomial you obtain when you sum 300th powers. (I didn't make that plot; it's from Richard Stanley's web page on interesting zeros of polynomials.)

It turns out that they're the Bernoulli polynomials; for very large n Veselov and Ward showed that the real zeroes are very near 0, ± 1/2, ± 1, ... if n is odd, and ± 1/4, ± 3/4, ± 5/4, ... if n is even; furthermore, in the limit, the nth Bernoulli polynomial has 2/(πe)n real zeros. (2/πe is about .235; thus in the 300th Bernoulli polynomial we expect about 70 real zeros, taking up an interval of length 35 or so centered at -1/2 on the real line; that's what you see in that plot.)

Goh and Boyer (who I've mentioned before for similar work on partition polynomials) have found the "zero attractor" of the Euler polynomials, and state in their paper that the methods there also give a similar result for the Bernoulli polynomials -- basically, what this means is that if we shrink down the plot of the zeros of the nth Bernoulli polynomial by a factor of n, then the zeroes fall very close to some limiting curves and are arranged with a certain density along those curves. (Along the portion of the real axis in question, the density is constant; along the other branches it doesn't seem to be.)

References:
William M. Y. Goh, Robert Boyer. On the Zero Attractor of the Euler Polynomials. arXiv: math.CO/0409062. (2004)
Alexander Veselov and Joseph Ward, On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials, arXiv: math.GM/0205183. (2002)

I've previously mentioned Richard Stanley's page of interesting zeros of polynomials. For an example of this in action, you can see a couple papers which were justed posted to the arXiv, by Robert P. Boyer and William M. Y. Goh:

Partition Polynomials: Asymptotics and Zeros

Polynomials associated with Partitions: Their Asymptotics and Zeros

In the first paper, we take the polynomial which counts the partitions of n by number of parts; when n = 200 the zeroes of this polynomial are plotted at Richard Stanley's web site. What's surprising is that the zeroes are actually a good bit more subtle than they'd look from that picture; they come in three families, the third of which doesn't appear until n = 13,000.

The second paper makes a similar conjecture for the zeroes of the "rank polynomial" and "crank polynomial" of high degree, where partitions are counted by their rank (the difference between the largest part and the number of parts) or the crank (see p. 11 for definition). (In the rank case, at least, Boyer and Goh actually just count partitions with positive rank; by conjugation one can get those with negative rank.) The zeroes of the rank polynomial appear to lie approximately on the unit circle, and are spread out uniformly.

By the way, I've previously credited this idea of looking at the zeros of combinatorially defined polynomials to Stanley. Apparently it's actually due to Rota, who once said:

The one contribution of mine that I hope will be remembered has consisted in just pointing out that all sorts of problems of combinatorics can be viewed as problems of location of the zeros of certain polynomials and in giving these zeros a combinatorial interpretation. This is now called the critical problem. Over the years people have added examples of critical problems, though we still haven't gotten any idea of how to solve it in general. I'd like to see someone get such an idea before I die. The four-color conjecture-that with only four colors you can color every planar map so that no two adjacent regions have the same color-is one of these critical problems.

Rota was Stanley's thesis advisor.