2014/08/23 by Ghioca, D., Nguyen, K. D. · 1 citation
#11G50 #37P15 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1408.5455
According to Medvedev and Scanlon, a polynomial f(x)∈ \mathbb Q[x] of degree d≥ 2 is called disintegrated if it is not linearly conjugate to xd or ± Cd(x) (where Cd(x) is the Chebyshev polynomial of degree d). Let n∈ℕ, let f1,…,fn∈ \mathbb Q[x] be disintegrated polynomials of degrees at least 2, and let φ=f1×…× fn be the corresponding coordinate-wise self-map of (\mathbb P1)n. Let X be an irreducible subvariety of (\mathbb P1)n of dimension r defined over \mathbb Q. We define the φ-anomalous locus of X which is related to the φ-periodic subvarieties of (\mathbb P1)n. We prove that the φ-anomalous locus of X is Zariski closed; this is a dynamical analogue of a theorem of Bombieri, Masser, and Zannier \citeBMZ07. We also prove that the points in the intersection of X with the union of all irreducible φ-periodic subvarieties of (\mathbb P1)n of codimension r have bounded height outside the φ-anomalous locus of X; this is a dynamical analogue of Habegger's theorem \citeHabegger09 which was previously conjectured in \citeBMZ07. The slightly more general self-maps φ=f1×…× fn where each fi∈ \mathbb Q(x) is a disintegrated rational map are also treated at the end of the paper.