2025/02/26 by Reis, Lucas
#11T55 (primary) and 37P05 (secondary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2502.19141
Let q be a power of a prime p, let \mathbb Fq be the finite field with q elements and, for each nonconstant polynomial F∈ \mathbb Fq[X] and each integer n≥ 1, let sF(n) be the degree of the splitting field (over \mathbb Fq) of the iterated polynomial F(n)(X). In 1999, Odoni proved that sA(n) grows linearly with respect to n if A∈ \mathbb Fq[X] is an additive polynomial not of the form aXph; moreover, if q=p and B(X)=Xp-X, he obtained the formula sB(n)=p\lceil logp n\rceil. In this paper we note that sF(n) grows at least linearly unless F∈ \mathbb Fq[X] has an exceptional form and we obtain a stronger form of Odoni's result, extending it to affine polynomials. In particular, we prove that if A is additive, then sA(n) resembles the step function p\lceil logp n\rceil and we indeed have the identity sA(n)=αp\lceil logp βn\rceil for some α, β∈ \mathbb Q, unless A presents a special irregularity of dynamical flavour. As applications of our main result, we obtain statistics for periodic points of linear maps over \mathbb Fqi as i→ +∞ and for the factorization of iterates of affine polynomials over finite fields.