Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limits of Mahler measures in multiple variables

François Brunault, Antonin Guilloux, Mahya Mehrabdollahei, Riccardo Pengo

TL;DR

This work extends the Lawton–Boyd framework to multivariate Laurent polynomials by showing that for any nonzero $P$ in $n$ variables and any sequence of integer matrices $A_d$ with $\rho(A_d)\to\infty$, the Mahler measures satisfy $\lim_{d\to\infty} m(P_{A_d})=m(P)$. It provides an explicit error bound generalizing the Dimitrov–Habegger result, relying on uniform $L^2$ bounds for $\log|P|$ and weak convergence of the push-forward measures $\mu_{A_d}$ to $\mu_n$, and it proves exponential convergence in the torus-nonvanishing case. The paper also develops a detailed analysis of the convergence speed, including an explicit bound and a full asymptotic expansion for a 2-variable family that converges to a 4-variable limit, with leading behavior $m(P_d)-m(P_\infty)\sim -\frac{\log\rho(A_d)}{2\rho(A_d)^2}$ and connections to Bernoulli numbers and zeta values. It discusses toric points, which can introduce quasi-periodic expansions and, in some cases, a logarithmic term, highlighting both the depth and limits of current techniques for describing convergence rates and asymptotics in multivariate Mahler measures.

Abstract

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

Limits of Mahler measures in multiple variables

TL;DR

This work extends the Lawton–Boyd framework to multivariate Laurent polynomials by showing that for any nonzero in variables and any sequence of integer matrices with , the Mahler measures satisfy . It provides an explicit error bound generalizing the Dimitrov–Habegger result, relying on uniform bounds for and weak convergence of the push-forward measures to , and it proves exponential convergence in the torus-nonvanishing case. The paper also develops a detailed analysis of the convergence speed, including an explicit bound and a full asymptotic expansion for a 2-variable family that converges to a 4-variable limit, with leading behavior and connections to Bernoulli numbers and zeta values. It discusses toric points, which can introduce quasi-periodic expansions and, in some cases, a logarithmic term, highlighting both the depth and limits of current techniques for describing convergence rates and asymptotics in multivariate Mahler measures.

Abstract

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.
Paper Structure (19 sections, 16 theorems, 111 equations, 4 figures)

This paper contains 19 sections, 16 theorems, 111 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Historical remarks
  3. Notation and conventions
  4. Generalities
  5. Matrices
  6. Measure theory
  7. Fourier coefficients
  8. Polynomials
  9. Constants
  10. A higher dimensional analogue of Lawton's theorem
  11. Convergence of measures and integrals
  12. Uniform L2-bounds and convergence of Mahler measures
  13. An error term in the convergence
  14. An explicit exponential convergence for polynomials without toric points
  15. An explicit error term in the general case
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every non-zero Laurent polynomial $P \in \mathbf C[z_1^{\pm 1}, \ldots, z_n^{\pm 1}] \setminus \{0\}$, and every sequence of integer $m\times n$-matrices $\{A_d\}_{d \in \mathbf N} \subseteq \mathbf Z^{m \times n}$ such that $\lim_{d \to +\infty} \rho(A_d) = +\infty$, we have that $\lim_{d \to +

Figures (4)

  • Figure 1: Plots of $m(P_{A_d}) - m(P)$, for $A_d = (1,d)$ and $200 \leq d \leq 1000$, which seem to lie on finitely many smooth branches. The SageMath code we used to produce these plots is available online Guilloux_2021.
  • Figure 2: Plots of $m(P_{A_d}) - m(P)$, for $A_d = (1,d)$ and $200 \leq d \leq 1000$, which seem to lie on infinitely many smooth branches. The SageMath code we used to produce these plots is available online Guilloux_2021.
  • Figure 3: Plot of $m(P_d) - m(P_\infty)$, for $200 \leq d \leq 1000$. The SageMath code we used to produce this plot is available online Guilloux_2021.
  • Figure 4: Plots of $m(P_{A_d}) - m(P)$, for $P = z_1^2 + z_2 + 1$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4: Dimitrov & Habegger
  • proof
  • proof : Proof of \ref{['thm:Lawton_generalisation']}
  • Theorem 4.1
  • ...and 28 more