Galois groups in a family of dynatomic polynomials

Abstract

Abstract. For every nonconstant polynomial f ∈ Q[x], let Φ 4,f denote the fourth dynatomic polynomial of f . We determine here the structure of the Galois group and the degrees of the irreducible factors of Φ 4,f for every quadratic polynomial f . As an application we prove new results related to a uniform boundedness conjecture of Morton and Silverman. In particular we show that if f is a quadratic polynomial, then, for more than 39% of all primes p, f does not have a point of period four in Qp.

Moduli Spaces for Dynamical Systems with Portraits

“…The classical example is the dynatomic curve X dyn 1 (n) whose points classify pairs (c, α) such that α is a point of exact (or formal) period n for the map x 2 + c, or more generally x d + c. (See Example 13.1 for further information about dynatomic curves and their description as portrait moduli spaces.) The following papers are among those that investigate X dyn 1 (n): Bousch [5], Buff-Epstein-Koch [6], Buff-Lei [7], Douady-Hubbard [13,14], Doyle [16], Doyle et al [17], Doyle-Poonen [18], Gao [24], Gao-Ou [25], Krumm [34,35], Lau-Schleicher [36], Morton [48]. These papers study topics such as smoothness, irreducibility, genus, and gonality of X dyn 1 (n), as well as reduction mod p and specialization properties.…”