Abstract
where 3 is meromorphic. In this case one may even obtain entire solutions, e.g. f = sin z, g = cos z, ,B = tan (z/2). Gross also shows that for n> 2 there are no entire solutions of (1), while for n> 3 there are no meromorphic solutions. Now the equation w3+z3 =1 defines an algebraic function whose Riemann surface has genus 1, and there is accordingly a uniformization by elliptic functions. If (P(z) is the Weierstrass elliptic function with periods wi, W2 satisfying ((P')2 = 4V3 g2(P g3, g2, g3 constants, then (cf. [2, p. 227]) w, and W2 may be chosen so that g2= 0 g3==1. With this d'(z) we find that
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