Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser
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| Format: | Preprint |
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2023
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| _version_ | 1866929196030230528 |
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| author | Dasgupta, Samit |
| author_facet | Dasgupta, Samit |
| contents | Let $\mathscr{L}$ denote the $\mathbf{Q}$-vector space of logarithms of algebraic numbers. In this expository work, we provide an introduction to the study of ranks of matrices with coefficients in $\mathscr{L}$. We begin by considering a slightly different question, namely we present a proof of a weak form of Baker's Theorem. This states that a collection of elements of $\mathscr{L}$ that is linearly independent over $\mathbf{Q}$ is in fact linear independent over $\overline{\mathbf{Q}}$. Next we recall Schanuel's Conjecture and prove Ax's analogue of it over $\mathbf{C}((t))$.
We then consider arbitrary matrices with coefficients in $\mathscr{L}$ and state the Structural Rank Conjecture, which gives a conjecture for the rank of a general matrix with coefficients in $\mathscr{L}$. We prove the theorem of Waldschmidt and Masser, which provides a lower bound giving a partial result toward the Structural Rank Conjecture. We conclude by stating a new conjecture that we call the Matrix Coefficient Conjecture, which gives a necessary condition for a square matrix with coefficients in $\mathscr{L}$ to be singular. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_02037 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser Dasgupta, Samit Number Theory 11J81 Let $\mathscr{L}$ denote the $\mathbf{Q}$-vector space of logarithms of algebraic numbers. In this expository work, we provide an introduction to the study of ranks of matrices with coefficients in $\mathscr{L}$. We begin by considering a slightly different question, namely we present a proof of a weak form of Baker's Theorem. This states that a collection of elements of $\mathscr{L}$ that is linearly independent over $\mathbf{Q}$ is in fact linear independent over $\overline{\mathbf{Q}}$. Next we recall Schanuel's Conjecture and prove Ax's analogue of it over $\mathbf{C}((t))$. We then consider arbitrary matrices with coefficients in $\mathscr{L}$ and state the Structural Rank Conjecture, which gives a conjecture for the rank of a general matrix with coefficients in $\mathscr{L}$. We prove the theorem of Waldschmidt and Masser, which provides a lower bound giving a partial result toward the Structural Rank Conjecture. We conclude by stating a new conjecture that we call the Matrix Coefficient Conjecture, which gives a necessary condition for a square matrix with coefficients in $\mathscr{L}$ to be singular. |
| title | Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser |
| topic | Number Theory 11J81 |
| url | https://arxiv.org/abs/2303.02037 |