تقرير
$P_d$ polynomials and Variants of Chinburg's Conjectures
| العنوان: | $P_d$ polynomials and Variants of Chinburg's Conjectures |
|---|---|
| المؤلفون: | Bertin, Marie-José, Mehrabdollahei, Mahya |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11R06 |
| الوصف: | This article provides some solutions to Chinburg's conjectures by studying a sequence of the Mahler measure of multivariate exact polynomials. These conjectures assert that for every odd quadratic Dirichlet Character of conductor $f$, $\chi_{-f}=\left(\frac{-f}{.}\right)$, there exists a bivariate polynomial (or a rational polynomial in the weak version) whose Mahler measure is a rational multiple of $L'(\chi_{-f},-1)$. To obtain such solutions for the conjectures we study a polynomial family denoted by $P_d(x,y)$, whose Mahler measure has been fully studied in \cite{mehrabdollahei2021mahler}, and \cite{BGMP}. 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 the conjectures. Specifically, we prove that this family provides solutions for conductors $f=3,4,8,15,20,24$. Notably, $P_d$ polynomials provide intriguing examples where the Mahler measures are linked to $L'(\chi,-1)$ with $\chi$ 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 the generalized version of Chinburg's conjecture, $P_d$ polynomials provide solutions for conductors $5,7,9$. Comment: 44 Pages, 25 tables |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.20634 |
| رقم الأكسشن: | edsarx.2407.20634 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |