On the non-Archimedean metric Mahler measure
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2014
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| _version_ | 1866908292854317056 |
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| author | Fili, Paul Samuels, Charles L. |
| author_facet | Fili, Paul Samuels, Charles L. |
| contents | Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted $M_\infty$, and prove that $M_\infty(α) = 1$ if and only if $α$ is a root of unity. We further show that $M_\infty$ defines a projective height on $\bar{\mathbb Q}^\times/ \bar{\mathbb Q}^\times_\mathrm{tors}$ as a vector space over $\mathbb Q$. Finally, we demonstrate how to compute $M_\infty(α)$ when $α$ is a surd. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_5084 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | On the non-Archimedean metric Mahler measure Fili, Paul Samuels, Charles L. Number Theory 11R04, 11R09 (Primary) Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted $M_\infty$, and prove that $M_\infty(α) = 1$ if and only if $α$ is a root of unity. We further show that $M_\infty$ defines a projective height on $\bar{\mathbb Q}^\times/ \bar{\mathbb Q}^\times_\mathrm{tors}$ as a vector space over $\mathbb Q$. Finally, we demonstrate how to compute $M_\infty(α)$ when $α$ is a surd. |
| title | On the non-Archimedean metric Mahler measure |
| topic | Number Theory 11R04, 11R09 (Primary) |
| url | https://arxiv.org/abs/1408.5084 |