Symbolic Computation Group

I will discuss this problem: given rational functions f and g over a field K , determine whether there are nonconstant rational functions u and v over K such that u(f(x)) = v(g(x)) . An equivalent problem is to compute the intersection of two fields which lie between K and K(x) . This has been solved completely in case f and g are polynomials and K has characteristic zero, but it remains open in nearly all other cases. I will present new and old results, examples, and algorithms for this problem.