Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture

2012/09/16 by Joseph H. Silverman, Silverman, Joseph H.
Mathematics · #11G35 #11J97 #37P55 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary: 37P15 #Secondary: 11B50

paper · pdf · doi:10.48550/arxiv.1209.3491

openalex publication_date 2012/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A primitive prime divisor of an element a_n of a sequence (a₁,a₂,a₃,...) is a prime P that divides a_n, but does not divide a_m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a_n has no primitive prime divisors. Let f : X --> X be a self-morphism of a variety, let D be an effective divisor on X, and let P be a point of X, all defined over the algebraic closure of Q. We consider the Zsigmondy set Z(X,f,P,D) of the sequence defined by the arithmetic intersection of the f-orbit of P with D. Under various assumptions on X, f, D, and P, we use Vojta's conjecture with truncated counting function to prove that the set of points fⁿ(P) with n in Z(X,f,P,D) is not Zariski dense in X.

Citations

• Zur Theorie der Potenzreste
• An upper bound for the height for regular affine automorphisms of Aⁿ

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