Level curves of rational functions and unimodular points on rational curves.

We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry. [ABSTRACT FROM AUTHOR]