  Let K be a function field in one variable over an arbitrary field F. Given a
rational function f(z) in K(z) of degree at least two, the associated canonical
height on the projective line was defined by Call and Silverman. The
preperiodic points of f all have canonical height zero; conversely, if F is a
finite field, then every point of canonical height zero is preperiodic.
However, if F is an infinite field, then there may be non-preperiodic points of
canonical height zero. In this paper, we show that for polynomial f, such
points exist only if f is isotrivial. In fact, such K-rational points exist
only if f is defined over the constant field of K after a K-rational change of
coordinates.
