A Quantitative Version of Runge's Theorem on Diophantine Equations

(C1) ai,n = am,j = 0 for all non-zero i and j, (C2) ai,j = 0 for all pairs (i, j) satisfying ni+mj > mn, (C3) the sum of all monomials ai,jxy of F for which ni+mj = nm is a constant multiple of a power of an irreducible polynomial in Z[x, y]. We note that (C2) is a stronger condition than (C1). The reason that (C1) is included above will be made clear in the statement of Theorem 1. We will make reference to the following condition which, together with (C1), is stronger than (C2) and (C3): (C4) the algebraic function y = y(x) defined by F (x, y) = 0 has only one class of conjugate Puiseux expansions.