In two dimensions, the [Green's function][1] 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][2].
\\[ \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][3]. 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.  


  [1]: http://en.wikipedia.org/wiki/Green's_function
  [2]: http://en.wikipedia.org/wiki/Dirac_delta_function
  [3]: http://en.wikipedia.org/wiki/Poisson%2527s_equation