IGL projects S’22

root dynamics for polynomials under differentiation

Take a (more or less, random) polynomial \(P\) of degree \(N\), and differentiate it several times keeping track of the roots of those sequential derivatives, \(P,P’,P^{\prime\prime},\ldots\). Remarkably, the roots seem to form tracks, as if sliding towards the origin like waterdrops.

All roots of all derivatives of a random polynomial of degree 100 (with iid normal complex coefficients) in polar coordinates (arguments rescaled by \(\pi\)). One can see the tracks easily.

A conceptual explanation (a physicist’s proof) is quite easy: in the standard electrostatic interpretation, the zeros of the derivative of a polynomial are the stationary point of the Coulomb force field corresponding to the ensemble of point charges sitting at the roots of that polynomial, and near a root, the remainder of the charges feel like a homogenized (locally nearly linear) potential.
For a mathematical elaboration of that idea, see Hanin’s paper here.

The goal of this project is to investigate the collective behaviors in more details. If one can pair the roots of a polynomial and its derivative, with high probability, one can form directed forests out of the collective roots of the derivatives. What are the typical shapes of those forests? What are their local statistics?

A related class of problems: zeros of a polynomial located on the boundary of a smooth curve, approximating a smooth density. The derivative behave in a rather controlled way.

Here a collection of roots was chosen nearly equispaced on the unit circle, but not quite: the poition of \(k\)-th root is \(2\pi k/n+h(k/n)\), where \(h\) is some trigonometric polynomial of low degree and low \(L_\infty\) norm.

A useful heuristic was developed recently by S. Steinenbergen. Namely, he asserts (and provides some evidence, both numeric and theoretical) that the roots of the derivative at \(t\in\Gamma\) locally shift by (a real multiple of)
\[
\arctan(Hu/u),
\]
where \(u\) is the local approximated density supported by the (real) line \(\Gamma\), and \(H\) is the Hilbert transform of that density (i.e.
\[Hu(t):=\mbox{p.v.} \int_\Gamma \frac{u(z)}{t-z}dz.\]

If we apply the same heuristic in our situation (where the density is supported by some smooth curve \(\Gamma\) in the complex plane), we will see phenomena shown on the left.

We see that these roots follow some quite regular evolution. The smoothness of the density is preserved (until some effects reminiscent of viscous Burgers equation start to kick in).

We conjecture that the arctan formula remains true for any smooth curve. Of course, we should interpret \(\arctan(w)\) as a branch of \[1/2\log \frac{1+iw}{1-iw}.\]

  • Faculty leads: Yuliy Baryshnikov and Tomoyuki Shirai (Kyushu University, Japan)
  • Level: Intermediate
  • Skills: Calculus 3; familiarity with complex numbers or electrostatics. Coding: some confidence with Python, Julia or Mathematica

department’s deep structure

What is our department good at? The question is not as simple as it seems: the research group composition changes fast, and their impact on the global scale might be smaller of larger than it seems.
The goal of this project is two-fold: on one hand, to detect the intrinsic research clusters (i.e. groups of people working in close areas, talking to the same communities, publishing in the same journals) within the department.
On the other hand, to see how significant these clusters are in the context of their respective fields: which are strong, which are growing…

Here’s a zero order approximation of this project: a quick analysis of what fraction of the global output in different areas (as defined by the MSC) our department produces:

The right display shows the fraction of the world’s math output that folks at University of Illinois produced between 2013 and 2019. On the left, the total number of published works in each area across all institutions is shown.

The usual caveats comparing publication numbers in different areas apply: some publish at a much higher rate than others; the disparities somewhat reflected in the right table. Still, the tables below invite some deeper level of introspection.

  • Faculty leads: Yuliy Baryshnikov
  • Level: Intermediate
  • Skills: Interest in data analysis. Coding: some confidence with R, Python