Polynomial-exponential equations in two variables

We consider equations of the form F(x, y, alpha(x), beta(y)) = 0 in integer unknowns x and y, where F is a polynomial with complex coefficients, and alpha and beta are non-zero complex numbers. We prove, outside of some exceptional cases, that if such an equation has infinitely many solutions, then alpha and beta are algebraic. (C) 1997 Academic Press.