Functional graphs of families of quadratic polynomials

TL;DR

This paper investigates the structure of functional graphs created by multiple quadratic polynomials over finite fields, linking leaf distribution to elliptic curve properties and providing numerical insights into larger polynomial families.

Contribution

It introduces new results on leaf counts in graphs generated by multiple quadratic polynomials and connects these to elliptic curve distributions, supported by extensive numerical data.

Findings

Leaf counts relate to elliptic curve Frobenius traces

Distribution of leaves follows Sato-Tate distribution in certain cases

Numerical results suggest patterns for larger polynomial families

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.