On the heuristic of approximating polynomials over finite fields by random mappings

Abstract

The behavior of iterations of functions is frequently approximated by 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 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 different random mapping models. We prove that the distribution of preimage sizes 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 [Formula: see text]. We show numerically that the behavior of Chebyshev polynomials of degree [Formula: see text] over finite fields present a sharp contrast when compared to other polynomials in their respective classes.