An exact family of bivariate polynomials and Variants of Chinburg's Conjectures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910895718793216 |
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| author | Bertin, Marie-José Mehrabdollahei, Mahya |
| author_facet | Bertin, Marie-José Mehrabdollahei, Mahya |
| contents | This article provides some solutions to Chinburg's conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet Character of conductor $f$, $χ_{-f}=\left(\frac{-f}{.}\right)$, there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of $L'(χ_{-f},-1)$. To obtain such solutions for the conjectures we investigate a polynomial family denoted by $P_d(x,y)$, whose Mahler measure has been recently studied. We demonstrate that the Mahler measure of $P_d$ can be expressed as a linear combination of Dirichlet $L$-functions, which has the potential to generate solutions to Chinburg's conjectures. Specifically, we prove that this family provides solutions for conductors $f=3,4,8,15,20$, and $24$. Notably, $P_d$ polynomials also provide intriguing examples where the Mahler measures are linked to $L'(χ,-1)$ with $χ$ being an odd non-real primitive Dirichlet character. These examples inspired us to generalize Chinburg's conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version of Chinburg's conjecture, $P_d$ polynomials provide solutions for conductors $5,7$, and $9$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20634 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An exact family of bivariate polynomials and Variants of Chinburg's Conjectures Bertin, Marie-José Mehrabdollahei, Mahya Number Theory 11R06 This article provides some solutions to Chinburg's conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet Character of conductor $f$, $χ_{-f}=\left(\frac{-f}{.}\right)$, there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of $L'(χ_{-f},-1)$. To obtain such solutions for the conjectures we investigate a polynomial family denoted by $P_d(x,y)$, whose Mahler measure has been recently studied. We demonstrate that the Mahler measure of $P_d$ can be expressed as a linear combination of Dirichlet $L$-functions, which has the potential to generate solutions to Chinburg's conjectures. Specifically, we prove that this family provides solutions for conductors $f=3,4,8,15,20$, and $24$. Notably, $P_d$ polynomials also provide intriguing examples where the Mahler measures are linked to $L'(χ,-1)$ with $χ$ being an odd non-real primitive Dirichlet character. These examples inspired us to generalize Chinburg's conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version of Chinburg's conjecture, $P_d$ polynomials provide solutions for conductors $5,7$, and $9$. |
| title | An exact family of bivariate polynomials and Variants of Chinburg's Conjectures |
| topic | Number Theory 11R06 |
| url | https://arxiv.org/abs/2407.20634 |