Galois orbits of torsion points over polytopes near atoral sets
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arXiv
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| Format: | Preprint |
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2024
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| author | Lin, Chenying |
| author_facet | Lin, Chenying |
| contents | Given an essentially atoral Laurent polynomial $P$, we show an equidistribution theorem for the function $\operatorname{log}|P|$ on specific subsets of Galois orbits of torsion points of the $d$-dimensional algebraic torus $\mathbb{G}^d_m(\overline{\mathbb{Q}})$. The specific subsets under consideration are the preimages of $d$-dimensional polytopes within the hypercube $[0,1]^d$ under the cotropicalization map. This generalises an equidistribution theorem of V. Dimitrov and P. Habegger, who considered only all Galois orbits that correspond to the entire hypercube $[0,1]^d$. In addition, we provide an estimate for the convergence speed of this equidistribution, expressed as a negative power of the strictness degree. Our approach is to derive an alternative version of Koksma's inequality over polytopes.
As an application, we provide the convergence speed of heights on a sequence of projective points for a specific two-dimensional example, answering a question posed by R. Gualdi and M. Sombra. In the appendix, we present an algorithm to compute the explicit value of the power of the strictness degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois orbits of torsion points over polytopes near atoral sets Lin, Chenying Number Theory Combinatorics Primary 11J83, 11G50, Secondary 14G40, 37P30, 52B11 Given an essentially atoral Laurent polynomial $P$, we show an equidistribution theorem for the function $\operatorname{log}|P|$ on specific subsets of Galois orbits of torsion points of the $d$-dimensional algebraic torus $\mathbb{G}^d_m(\overline{\mathbb{Q}})$. The specific subsets under consideration are the preimages of $d$-dimensional polytopes within the hypercube $[0,1]^d$ under the cotropicalization map. This generalises an equidistribution theorem of V. Dimitrov and P. Habegger, who considered only all Galois orbits that correspond to the entire hypercube $[0,1]^d$. In addition, we provide an estimate for the convergence speed of this equidistribution, expressed as a negative power of the strictness degree. Our approach is to derive an alternative version of Koksma's inequality over polytopes. As an application, we provide the convergence speed of heights on a sequence of projective points for a specific two-dimensional example, answering a question posed by R. Gualdi and M. Sombra. In the appendix, we present an algorithm to compute the explicit value of the power of the strictness degree. |
| title | Galois orbits of torsion points over polytopes near atoral sets |
| topic | Number Theory Combinatorics Primary 11J83, 11G50, Secondary 14G40, 37P30, 52B11 |
| url | https://arxiv.org/abs/2412.11156 |