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

Marie-José Bertin, Mahya Mehrabdollahei

TL;DR

This work builds a direct bridge between multivariate Mahler measures and special values of Dirichlet $L$-functions via the polynomial family $\{P_d\}_d$. By expressing $m(P_d)$ as a linear combination of $L'({\chi},-1)$ through the dilogarithm at roots of unity, the authors construct explicit product- and ratio-based Mahler-measure identities that realize partial solutions to Chinburg's conjectures for conductors $f=3,4,8,15,20,24$, and extend to a generalized weak version for all primitive odd characters. The results hinge on exactness of $P_d$, the tempered structure of its Newton polygon, and the Bloch-Wigner dilogarithm, together with Zagier's and Möbius-based linear-algebra decompositions that place coefficients in cyclotomic fields. These findings not only recover known cases (e.g., Smyth, Ray) but also provide a scalable framework to generate further $R_f$ or $Q_f$-type objects achieving $L'({\chi},-1)$ values, potentially advancing the landscape of Chinburg-type conjectures. The work suggests directions for algorithmic generation of solutions, exploration of additional exact polynomial families, and deeper links between Mahler measures, dilogarithms, and Dirichlet $L$-functions with practical implications for number theory and algebraic $K$-theory.

Abstract

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 $9$.

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

TL;DR

This work builds a direct bridge between multivariate Mahler measures and special values of Dirichlet -functions via the polynomial family . By expressing as a linear combination of through the dilogarithm at roots of unity, the authors construct explicit product- and ratio-based Mahler-measure identities that realize partial solutions to Chinburg's conjectures for conductors , and extend to a generalized weak version for all primitive odd characters. The results hinge on exactness of , the tempered structure of its Newton polygon, and the Bloch-Wigner dilogarithm, together with Zagier's and Möbius-based linear-algebra decompositions that place coefficients in cyclotomic fields. These findings not only recover known cases (e.g., Smyth, Ray) but also provide a scalable framework to generate further or -type objects achieving values, potentially advancing the landscape of Chinburg-type conjectures. The work suggests directions for algorithmic generation of solutions, exploration of additional exact polynomial families, and deeper links between Mahler measures, dilogarithms, and Dirichlet -functions with practical implications for number theory and algebraic -theory.

Abstract

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 , , there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of . To obtain such solutions for the conjectures we investigate a polynomial family denoted by , whose Mahler measure has been recently studied. We demonstrate that the Mahler measure of can be expressed as a linear combination of Dirichlet -functions, which has the potential to generate solutions to Chinburg's conjectures. Specifically, we prove that this family provides solutions for conductors , and . Notably, polynomials also provide intriguing examples where the Mahler measures are linked to with 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, polynomials provide solutions for conductors , and .
Paper Structure (10 sections, 11 theorems, 59 equations, 7 tables)

This paper contains 10 sections, 11 theorems, 59 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Historical remarks
  3. The Mahler measure of family and links with -functions
  4. A brief introduction to family
  5. Representation of in terms of -functions
  6. Solutions to Chinburg's conjectures using the family
  7. Perspective
  8. Appendix
  9. Conrey label of Dirichlet characters used in LMFDB
  10. Tables

Key Result

Theorem 1.3

Let $d\in \mathbb{Z}_{\geq 1}$, and for every primitive odd Dirichlet character $\chi$ of conductor $k$, such that $k$ divides $(d+1)(d+2)$, there exists a coefficient $C_{k,\chi}^d\in \mathbb{Q}\left(e^{\frac{2\pi i}{\phi(k)}}\right)\subset \mathbb{Q}\left(e^{\frac{2\pi i}{\phi((d+1)(d+2))}}\right)

Theorems & Definitions (34)

  • Definition 1.1: K. Mahler, 11
  • Definition 1.2
  • Theorem 1.3
  • Conjecture 1.4: Strong Chinburg's Conjecture chinburge
  • Conjecture 1.5: Weak Chinburg's Conjecture chinburge
  • Conjecture 1.6: Generalization of the weak Chinburg's Conjecture
  • Conjecture 1.7: Generalization of the strong Chinburg's Conjecture
  • Proposition 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 24 more