An exact family of bivariate polynomials and variants of Chinburg’s Conjectures

This article provides partial 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=( -f/.) χ - f = - f . , 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) L ′ ( χ - f , - 1 ) . We prove that the Mahler measure of a polynomial family, denoted by Pd P d , can be expressed as a linear combination of the derivatives of Dirichlet L -functions. Specifically, this family provides solutions to the conjectures for conductors $$f=3,4,8,15,20$$ f = 3 , 4 , 8 , 15 , 20 , and 24. We further generalize Chinburg’s conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version, the polynomials Pd P d provide solutions for conductors 5, 7, and 9.