Pd polynomials and Variants of Chinburg's Conjectures

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, χ-f=(-f/.), there exists a bivariate polynomial (or a rational polynomial in the weak version) whose Mahler measure is a rational multiple of L'(χ-f,-1). To obtain such solutions for the conjectures we study a polynomial family denoted by Pd(x,y), whose Mahler measure has been fully studied in {mehrabdollahei2021mahler}, and {BGMP}. We demonstrate that the Mahler measure of Pd 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, Pd polynomials 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 the generalized version of Chinburg's conjecture, Pd polynomials provide solutions for conductors $5,7,9$.