An exact family of bivariate polynomials and Variants of Chinburg's Conjectures

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 $f$, ${\chi }_{-f}=\left(\frac{-f}{\mathrm{.}}\right)$, there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of $L^{\mathrm{\prime }}({\chi }_{-f},-1)$. 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 $L^{\mathrm{\prime }}(\chi ,-1)$ 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 this generalized version of Chinburg’s conjecture, $P_d$ polynomials provide solutions for conductors $5,7$, and $9$.