On a uniform bound for the number of exceptional linear subvarieties in the dynamical Mordell-Lang conjecture

TL;DR

This paper establishes a uniform bound on the number of exceptional linear subvarieties intersecting orbits in the dynamical Mordell-Lang conjecture, specifically for the d'th-power map with multiplicatively independent coordinates.

Contribution

It proves a uniform bound on the number of special linear subvarieties related to orbits under the d'th-power map, independent of the point or degree.

Findings

Finitely many super-spanned linear subvarieties exist for the d'th-power map.

The number of such subvarieties is bounded solely by the dimension n.

A finite subset S ensures points outside S are in linear general position.

Abstract

Let F : P^n --> P^n be a morphism of degree d > 1 defined over C. The dynamical Mordell--Lang conjecture says that the intersection of an orbit O_F(P) and a subvariety X of P^n is usually finite. We consider the number of linear subvarieties L in P^n such that the intersection of O_F(P)and L is "larger than expected." When F is the d'th-power map and the coordinates of P are multiplicatively independent, we prove that there are only finitely many linear subvarieties that are "super-spanned" by O_F(P), and further that the number of such subvarieties is bounded by a function of n, independent of the point P or the degree d. More generally, we show that there exists a finite subset S, whose cardinality is bounded in terms of n, such that any n+1 points in O_F(P)-S are in linear general position in P^n.