On the Length of Critical Orbits of Stable Quadratic Polynomials

TL;DR

This paper establishes an improved upper bound of O(q^{3/4}) for the length of critical orbits of stable quadratic polynomials over finite fields, using advanced bounds on character sums and recent mathematical results.

Contribution

It introduces a novel application of Weil bounds and recent research to significantly refine the known bounds on critical orbit lengths in finite fields.

Findings

Critical orbit length is bounded by O(q^{3/4})

Uses Weil bounds of character sums

Builds on recent results by Boston and Jones

Abstract

We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of $q$ elements is of length $O(q^{3/4})$, improving upon the trivial bound $q$.