## Involve: A Journal of Mathematics

- Involve
- Volume 11, Number 1 (2018), 169-179.

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

Elisa Bellah, Derek Garton, Erin Tannenbaum, and Noah Walton

#### Abstract

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.

#### Article information

**Source**

Involve, Volume 11, Number 1 (2018), 169-179.

**Dates**

Received: 17 October 2016

Accepted: 5 December 2016

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513775049

**Digital Object Identifier**

doi:10.2140/involve.2018.11.169

**Mathematical Reviews number (MathSciNet)**

MR3681355

**Zentralblatt MATH identifier**

06762712

**Subjects**

Primary: 37P05: Polynomial and rational maps

Secondary: 05C80: Random graphs [See also 60B20] 37P25: Finite ground fields

**Keywords**

arithmetic dynamics functional graphs finite fields polynomials rational maps

#### Citation

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