  The study of iterations of functions over a finite field and the
corresponding functional graphs is a growing area of research with connections
to cryptography. The behaviour of such iterations is frequently approximated by
what is know as the Brent-Pollard heuristic, where one treats functions as
random mappings. We aim at understanding this heuristic and focus on the
expected rho length of a node of the functional graph of a polynomial over a
finite field. Since the distribution of indegrees (preimage sizes) of a class
of functions appears to play a central role in its average rho length, we
survey the known results for polynomials over finite fields giving new proofs
and improving one of the cases for quartic polynomials. We discuss the
effectiveness of the heuristic for many classes of polynomials by comparing our
experimental results with the known estimates for random mapping models defined
by different restrictions on their distribution of indegrees. We prove that the
distribution of indegrees of general polynomials and mappings have similar
asymptotic properties, including the same asymptotic average coalescence. The
combination of these results and our experiments suggests that these
polynomials behave like random mappings, extending a heuristic that was known
only for degree $2$. We show numerically that the behaviour of Chebyshev
polynomials of degree $d \geq 2$ over finite fields present a sharp contrast
when compared to other polynomials in their respective classes.
