Algebraic curves and meromorphic functions sharing pairs of values

Abstract

The 4IM+1CM-problem is to determine all pairs \((f,g)\) of meromorphic functions in the complex plane that are not Möbius transformations of each other and share five pairs of values, one of them CM (counting multiplicities). In the present paper it is shown that each such pair parameterises some algebraic curve \(K(x,y)=0\) of genus zero and low degree. Thus the search may be restricted to the pairs of meromorphic functions \((Q(e^z),\widetilde Q(e^z))\), where \(Q\) and \(\widetilde Q\) are non-constant rational functions of low degree. This leads to the paradoxical situation that the 4IM+1CM-problem could be solved by a computer algebra virtuoso rather than a complex analyst.