Primitive prime divisors in the forward orbit of a polynomial

2025/01/22 by Laishram, Shanta, Rout, Sudhansu S., Yadav, Prabhakar
#11B37 (Primary) #37F10 #37P05 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.02600

Abstract

For the polynomial f(z) ∈ ℚ[z], we consider the Zsigmondy set Z(f,0) associated to the numerators of the sequence \fⁿ(0)\_{n ≥ 0}. In this paper, we provide an upper bound on the largest element of Z(f, 0). As an application, we show that the largest element of the set Z(f,0) is bounded above by 6 when f(z) = z^d + z^e +c ∈ ℚ[z], with d>e ≥ 2 and |c|>2. Furthermore, when f(z) =z^d+c ∈ ℚ[z] with |f(0)| > 2^(d)/(d-1) and d >2, we also deduce a result of Krieger [Int. Math. Res. Not. IMRN, 23 (2013), pp. 5498-5525] as a consequence of our main result.

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• Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials
• Primitive prime divisors in the critical orbit of z^d+c
• Prime divisors in polynomial orbits over function fields
• Primitive prime divisors in zero orbits of polynomials
• Common divisors of totients of polynomial sequences