  Let U be the open unit disc in C. Given a continuous function g: bU --> C-{0}
denote by W(g) the winding number of g around the origin. We prove that a
continuous function f: bU --> C extends meromorphically through U if and only
if there is a nonnegative integer N such that W(Pf+Q) is greater than or equal
to -N for every pair P,Q of polynomials such that Pf+Q has no zero on bU. If
this is the case then the meromorphic extension of f has at most N poles in U,
counting multiplicity.
