Optimal factorizations of rational numbers using factorization trees
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2014
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| _version_ | 1866912302450606080 |
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| author | Samuels, Charles L. Strunk, Tanner J. |
| author_facet | Samuels, Charles L. Strunk, Tanner J. |
| contents | Let $m_t(α)$ denote the $t$-metric Mahler measure of the algebraic number $α$. Recent work of the first author established that the infimum in $m_t(α)$ is attained by a single point $\barα= (α_1,\ldots,α_N)\in \overline{\mathbb Q}^N$ for all sufficiently large $t$. Nevertheless, no efficient method for locating $\bar α$ is known. In this article, we define a new tree data structure, called a factorization tree, which enables us to find $\barα$ when $α\in \mathbb Q$. We establish several basic properties of factorization trees, and use these properties to locate $\barα$ in previously unknown cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1408_4162 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Optimal factorizations of rational numbers using factorization trees Samuels, Charles L. Strunk, Tanner J. Number Theory 11A51, 11G50, 11R09 (Primary), 05C05, 05C20, 68P05 (Secondary)} Let $m_t(α)$ denote the $t$-metric Mahler measure of the algebraic number $α$. Recent work of the first author established that the infimum in $m_t(α)$ is attained by a single point $\barα= (α_1,\ldots,α_N)\in \overline{\mathbb Q}^N$ for all sufficiently large $t$. Nevertheless, no efficient method for locating $\bar α$ is known. In this article, we define a new tree data structure, called a factorization tree, which enables us to find $\barα$ when $α\in \mathbb Q$. We establish several basic properties of factorization trees, and use these properties to locate $\barα$ in previously unknown cases. |
| title | Optimal factorizations of rational numbers using factorization trees |
| topic | Number Theory 11A51, 11G50, 11R09 (Primary), 05C05, 05C20, 68P05 (Secondary)} |
| url | https://arxiv.org/abs/1408.4162 |