Stability of certain higher degree polynomials.

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. A polynomial f (z) ∈ ℚ [ z ] is stable over ℚ if irreducibility of f (z) implies that all its iterates are also irreducible over ℚ , that is, f n (z) is irreducible over ℚ for all n ≥ 1 , where f n (z) denotes the n -fold composition of f (z). In this paper, we study the stability of f (z) = z d + 1 c for d ≥ 2 , c ∈ ℤ ∖ { 0 }. We show that for infinite families of d ≥ 3 , whenever f (z) is irreducible, all its iterates are irreducible, that is, f (z) is stable. Under the assumption of explicit a b c -conjecture, we further prove the stability of f (z) = z d + 1 c for the remaining values of d. Also for d = 3 , if f (z) is reducible, then the number of irreducible factors of each iterate of f (z) is exactly 2 for | c | ≤ 1 0 1 2 . [ABSTRACT FROM AUTHOR]