Functional graphs of families of quadratic polynomials

Mans, Bernard; Sha, Min; Shparlinski, Igor; Sutantyo, Daniel

doi:10.1090/mcom/3838

Functional graphs of families of quadratic polynomials
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by Bernard Mans, Min Sha, Igor E. Shparlinski and Daniel Sutantyo HTML | PDF

Math. Comp. 92 (2023), 2307-2331 Request permission

Abstract:

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.

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