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A probabilistic
heuristic for counting components of functional graphs of
polynomials over finite fields
Elisa Bellah, Derek Garton, Erin Tannenbaum and Noah Walton
Vol. 11 (2018), No. 1, 169–179
DOI: 10.2140/involve.2018.11.169
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Abstract |
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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. |
Keywords
arithmetic dynamics, functional graphs, finite fields,
polynomials, rational maps
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Mathematical Subject Classification 2010
Primary: 37P05
Secondary: 05C80, 37P25
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Milestones
Received: 17 October 2016
Accepted: 5 December 2016
Published: 17 July 2017
Communicated by Michael E. Zieve
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Authors |
| Elisa Bellah | |
| Department of Mathematics University of Oregon Eugene, OR 97403 United States |
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| Derek Garton | |
| Fariborz Maseeh Department of
Mathematics and Statistics Portland State University PO Box 751 Portland, OR 97207 United States |
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| Erin Tannenbaum | |
| Fariborz Maseeh Department of
Mathematics and Statistics Portland State University PO Box 751 Portland, OR 97207 United States |
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| Noah Walton | |
| Fariborz Maseeh Department of
Mathematics and Statistics Portland State University PO Box 751 Portland, OR 97207 United States |
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