Likely intersections in powers of the multiplicative group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916802046459904 |
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| author | Dill, Gabriel Andreas Gallinaro, Francesco |
| author_facet | Dill, Gabriel Andreas Gallinaro, Francesco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07550 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Likely intersections in powers of the multiplicative group Dill, Gabriel Andreas Gallinaro, Francesco Number Theory Algebraic Geometry Logic 14L10, 03C98, 11U09, 14T90 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. |
| title | Likely intersections in powers of the multiplicative group |
| topic | Number Theory Algebraic Geometry Logic 14L10, 03C98, 11U09, 14T90 |
| url | https://arxiv.org/abs/2506.07550 |