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

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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


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