We determine all the complex polynomials f ðX Þ such that, for two suitable distinct, nonconstant rational functions gðtÞ and hðtÞ, the equality f ðgðtÞÞ ¼ f ðhðtÞÞ holds. This extends former results of Tverberg, and is a contribution to the more general question of determining the polynomials f ðX Þ over a number field K such that f ðX Þ À l has at least two distinct K-rational roots for infinitely many l 2 K.