Question

In two dimensions, the Green's function of the Laplacian is the natural logarithm, $\nabla^2 \ln|z| = \delta(z)$, so we can take log of a polynomial the sum of delta-functions. \[ \nabla^2 \ln p(z) = \sum \delta(z - z_i) \] where $z_i$ runs over the roots of $p(z)=0$.

The equation $\nabla^2 \phi = \rho$ is Poisson's equation. In our case, the charge distribution is the sum of point charges. I wonder if anyone has studied roots of polynomial equations by analogy to Electrostatics.

Answer 1

For certain orthogonal polynomials, there is an electrostatic interpretation that goes back to Stieltjes. For example, suppose we put charges of $(1+\alpha)/2$ and $(1+\beta)/2$ at $1$ and $-1$, respectively, and put $n$ particles with charge $1$ in between them. Then the minimal energy configuration of the $n$ particles is at the roots of the $n$-th degree Jacobi polynomial with parameters $(\alpha,\beta)$. See http://arxiv.org/abs/math/0512293 for a survey of related results.