(The published version and the arXiv version are available online.)

We find all polynomials  f, g, h  over a field  K  such that  g  and  h  are linear and  f(g(x)) = h(f(x)). We also solve the same problem for rational functions  f, g, h  in case the field  K  is algebraically closed.

Special cases of the polynomial result were proved previously by Wells, Mullen, Park, and Eigenthaler–Nöbauer. These problems were solved long ago in case  K  has characteristic zero, but they are more difficult in positive characteristic. More generally, although the commuting rational functions in characteristic zero have been known since the 1920's, little is known about the analogous problem in positive characteristic. We view our result as a first step in this direction.

Michael Zieve:  home page   publication list