Irreducibility of polynomials defining parabolic parameters of period 3
Volume 221 / 2025
Acta Arithmetica 221 (2025), 253-270
MSC: Primary 11R09; Secondary 11R04
DOI: 10.4064/aa241008-26-9
Published online: 7 November 2025
Abstract
Morton and Vivaldi defined the polynomials whose roots are parabolic parameters for a one-parameter family of polynomial maps (called delta factors here). 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 more, this becomes a challenging problem. We prove the irreducibility of delta factors for period $3$ and demonstrate the existence of infinitely many irreducible delta factors for periods greater than $3$.
