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arXiv · 2506.07550

Likely intersections in powers of the multiplicative group

Gabriel Andreas Dill·Francesco Gallinaro
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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.

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Gabriel Andreas Dill, Francesco Gallinaro. 2025-06-09. Likely intersections in powers of the multiplicative group. https://arxiv.org/abs/2506.07550

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