Irreducibility of polynomials defining parabolic parameters of period 3

TL;DR

This paper proves the irreducibility of polynomials associated with parabolic parameters of period 3 in quadratic maps and shows infinitely many such irreducible polynomials exist for higher periods.

Contribution

It establishes the irreducibility of delta factors for period 3 and proves the existence of infinitely many irreducible delta factors for periods greater than 3.

Findings

Proved irreducibility of delta factors for period 3

Established existence of infinitely many irreducible delta factors for higher periods

Extended understanding of polynomial irreducibility in dynamical systems

Abstract

Morton and Vivaldi defined the polynomials whose roots are parabolic parameters for a one-parameter family of polynomial maps. We call these polynomials delta factors. They conjectured that delta factors are irreducible for the family $z\mapsto z^2+c$. One can easily show the irreducibility for periods $1$ and $2$ by reducing it to the irreducibility of cyclotomic polynomials. However, for periods $3$ and beyond, this becomes a challenging problem. This paper proves the irreducibility of delta factors for the period $3$ and demonstrates the existence of infinitely many irreducible delta factors for periods greater than $3$.