Likely intersections in powers of the multiplicative group

TL;DR

This paper establishes finiteness properties of intersections between algebraic subvarieties of powers of the multiplicative group and translates of subtori, using tropical geometry, equidistribution, and model theory.

Contribution

It introduces new finiteness results on intersections of subvarieties with translates of subtori, based on geometrical non-degeneracy, expanding understanding in algebraic geometry and number theory.

Findings

Every translate of a subtorus intersects the variety unless contained in finitely many proper subtori.

Every translate by a torsion point intersects the variety unless contained in finitely many proper algebraic subgroups.

Methods combine tropical geometry, equidistribution, and model theory.

Abstract

We derive two finiteness properties as consequences of the geometrical non-degeneracy of an algebraic subvariety $W$ of a power of the multiplicative group, concerning the intersections of $W$ with translates of a subtorus $H$ of dimension greater than or equal to the codimension of $W$. The first one is that every translate of $H$ intersects $W$, unless $H$ is contained in one of finitely many proper subtori depending only on $W$. The second one is that every translate of $H$ by a torsion point intersects $W$, unless the translate is contained in one of finitely many proper algebraic subgroups, again depending only on $W$. We use methods from tropical geometry and equidistribution, as well as some very mild model theory.