  For certain polynomials we relate the number of roots inside the unit circle
with the index of a non-degenerate isolated umbilic point on a real analytic
surface in Euclidean 3-space.
  In particular, for $N>0$ we prove that for a certain ($N+2$)-real dimensional
family of complex polynomials of degree $N$, the number of roots inside the
unit circle is less than or equal to $1+N/2$. This bound is established as
follows.
  From the polynomial we construct a convex real analytic surface containing an
isolated umbilic point, such that the index of the umbilic point is determined
by the number of roots of the polynomial that lie inside the unit circle. The
bound on the number of roots then follows from Hamburger's bound on the index
of an isolated umbilic point on a convex real analytic surface.
  The class of polynomials that arise are those with self-inversive second
derivative. Thus the number of roots inside the unit circle is proven to be
bounded for a polynomial with self-inversive second derivative.
