Motion of zeroes of complex polynomials

Consider the polynomial f(z) = (z-1)(z-2)...(z-20). Clearly it has 20 roots; these are 1, 2, ..., 20.

Now consider the polynomial g(z) = -z²⁰. It also has 20 roots, namely the origin with multiplicity 20.

And consider h(z) = t f(z) + (1-t) g(z), as t varies from 0 to 1. (Most of the "action" happens when t is very near 0 or 1, so this probably isn't the best parametrization.) Now, as t varies, you can find the roots numerically. Imagine the roots as twenty particles moving around in the plane. What happens, basically, is that as t increases roots start by sliding along the real axis, towards each other in pairs -- this is what you expect if you just plot f(z) as a real polynomial. (Interestingly, the collision appears to be perfectly elastic.) They then bang into each other and head off in the positive and negative imaginary directions. And eventually they curve around and approach the origin, on paths spaced 18 degrees apart. (I can try to produce graphics.)

There's nothing special about these polynomials -- that is, I suspect that something like this happens more generally. This is actually just an extension of an example in Peter Henrici's Applied and computational complex analysis (volume 1, p. 282) -- Henrici says that J. H. Wilkinson looked at the polynomial f(z) - 2^-23z¹⁹ and saw that it had five pairs of complex conjugate zeroes.

But it seems like there should be some sort of general theory of the way that roots of families of polynomials "move around" in the plane. (And if there isn't, why not?) Does the situation I've described ring a bell for anybody?

Zeros of some polynomials arising from sums

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)