On functional graphs of quadratic polynomials

Bernard Mans, Min Sha*, Igor E. Shparlinski, Daniel Sutantyo

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These include: the number of connected functional graphs, the number of graphs having a maximal cycle, the number of cycles of fixed size, the number of components of fixed size, as well as the shape of trees extracted from functional graphs. We particularly focus on connected functional graphs, that is, the graphs that contain only one component (and thus only one cycle). Based on the results of our computations, we formulate several conjectures highlighting the similarities and differences between these functional graphs and random mappings.

Original languageEnglish
Pages (from-to)292-300
Number of pages9
JournalExperimental Mathematics
Volume28
Issue number3
Early online date29 Nov 2017
DOIs
Publication statusPublished - 3 Jul 2019

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Keywords

  • algorithms
  • finite fields
  • functional graphs
  • Polynomial maps
  • random maps

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