  We consider the zeros of the sum of independent random polynomials as their
degrees tend to infinity. Namely, let $p$ and $q$ be two independent random
polynomials of degree $n$, whose roots are chosen independently from the
probability measures $\mu$ and $\nu$ in the complex plane, respectively. We
compute the limiting distribution for the zeros of the sum $p+q$ as $n$ tends
to infinity. The limiting distribution can be described by its logarithmic
potential, which we show is the pointwise maximum of the logarithmic potentials
of $\mu$ and $\nu$. More generally, we consider the sum of $m$ independent
degree $n$ random polynomials when $m$ is fixed and $n$ tends to infinity. Our
results can be viewed as describing a version of the free additive convolution
from free probability theory for zeros of polynomials.
