F-sets and finite automata

TL;DR

This paper extends the concept of automatic sets to finitely generated abelian groups with endomorphisms, applying it to the Mordell-Lang problem in positive characteristic and generalizing classical theorems.

Contribution

It introduces F-automatic subsets in abelian groups with endomorphisms and applies this to the Mordell-Lang problem, generalizing Skolem-Mahler-Lech theorem in this context.

Findings

F-subsets are F-automatic in the given setting.

X intersect Γ is F-automatic for semiabelian G and closed subvariety X.

F-subsets are F-normal, extending Derksen's notions.

Abstract

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $\Gamma$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $\Gamma$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $\Gamma$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subvariety then X intersect $\Gamma$ is F-automatic. Derksen's notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F-subsets are F-normal. In particular, the X intersect $\Gamma$ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.