On Rational Functions with More than Three Branch Points

摘要

Let d be a positive integer and A be a collection of partitions of d of the form (a(1), ..., a(p)), (b(1), ..., b(q)), (m(1) + 1, 1, ..., 1), ..., (m(l) + 1, 1, ..., 1) , where (m(1), ..., m(l)) is a partition of p + q - 2 > 0. We prove that there exists a rational function on the Riemann sphere with branch data Lambda if and only if max(m(1), ..., m(l)) < d/GCD(a1, ..., a(p), b(1), ..., b(q)). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.