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.