Involve: A Journal of Mathematics
- Volume 11, Number 1 (2018), 169-179.
A probabilistic heuristic for counting components of functional graphs of polynomials over finite fields
Flynn and Garton (2014) 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 Kruskal’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.
Involve, Volume 11, Number 1 (2018), 169-179.
Received: 17 October 2016
Accepted: 5 December 2016
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Bellah, Elisa; Garton, Derek; Tannenbaum, Erin; Walton, Noah. A probabilistic heuristic for counting components of functional graphs of polynomials over finite fields. Involve 11 (2018), no. 1, 169--179. doi:10.2140/involve.2018.11.169. https://projecteuclid.org/euclid.involve/1513775049