A probabilistic heuristic for counting components of functional graphs of polynomials over finite fields

TL;DR

This paper introduces a heuristic to estimate the number of components in functional graphs of polynomials over finite fields, bridging the gap between previous bounds and providing near-optimal estimates for small degrees.

Contribution

The authors propose a novel heuristic that approximates cycle counts in polynomial functional graphs, leading to bounds close to Kruskal's asymptotic for all degrees.

Findings

Heuristic implies average component count is within a constant factor of Kruskal's bound.

Numerical data supports the heuristic's accuracy in estimating component counts.

Analysis extends previous bounds to small degree polynomials.

Abstract

In 2014, Flynn and the second author bounded the average number of components of the functional graphs of polynomials of fixed degree over a finite field. When the fixed degree was large (relative to the size of the finite field), their lower bound matched Kruskal's asymptotic for random functional graphs. However, when the fixed degree was small, they were unable to match Krusal's bound, since they could not (Lagrange) interpolate cycles in functional graphs of length greater than the fixed degree. In our work, we introduce a heuristic for approximating the average number of such cycles of any length. This heuristic is, roughly, that for sets of edges in a functional graph, the quality of being a cycle and the quality of being interpolable are "uncorrelated enough". We prove that this heuristic implies that the average number of components of the functional graphs of polynomials of fixed degree over a finite field is within a bounded constant of Kruskal's bound. We also analyze some numerical data comparing implications of this heuristic to some component counts of functional graphs of polynomials over finite fields.