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Random walks through the areal Mahler measure: steps in the complex plane

Matilde N. Lalín, Siva Sankar Nair, Berend Ringeling, Subham Roy

TL;DR

This work establishes explicit formulas for the areal Mahler measure $m_{\mathbb{D}}$ for two fundamental polynomial families: $x+y+k$ and $Q_k=(x+1)(y+1)+kz$. By interpreting $m_{\mathbb{D}}$ through the areal Zeta Mahler function and a random-walk framework, the authors derive hypergeometric, dilogarithmic, and modular-analytic expressions that relate $m_{\mathbb{D}}$ to the classical Mahler measure $m$, plus contributions from Deninger-cycle volumes. They present two proofs for the main results in the $x+y+k$ case (a hypergeometric/ZAMM approach and a direct computation), and they extend the methodology to $Q_k$, obtaining density formulas and a detailed expansion of $m_{\mathbb{D}}(Q_k)$ that includes a central volume term $c_0(k)$ with modular significance. The results yield efficient numerical schemes, illuminate connections to special values of $L$-functions and Bloch–Wigner dilogarithms, and reveal geometric (Deninger-cycle) and modular structures underlying areal Mahler measures.

Abstract

We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.

Random walks through the areal Mahler measure: steps in the complex plane

TL;DR

This work establishes explicit formulas for the areal Mahler measure for two fundamental polynomial families: and . By interpreting through the areal Zeta Mahler function and a random-walk framework, the authors derive hypergeometric, dilogarithmic, and modular-analytic expressions that relate to the classical Mahler measure , plus contributions from Deninger-cycle volumes. They present two proofs for the main results in the case (a hypergeometric/ZAMM approach and a direct computation), and they extend the methodology to , obtaining density formulas and a detailed expansion of that includes a central volume term with modular significance. The results yield efficient numerical schemes, illuminate connections to special values of -functions and Bloch–Wigner dilogarithms, and reveal geometric (Deninger-cycle) and modular structures underlying areal Mahler measures.

Abstract

We study the areal Mahler measure of the two-variable, -parameter family and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.

Paper Structure

This paper contains 14 sections, 37 theorems, 231 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The areal Mahler measure of $x+y+k$
  4. The areal Zeta Mahler function of $x + y+k$
  5. The areal Mahler measure of $x + y+k$
  6. The Mahler measure of $x + y+k$ and the Proof of Theorem \ref{['thm:x+y+k']}
  7. A direct computation for the areal Mahler measure of $x+y+k$
  8. The areal Mahler measure of $(x+1)(y+1)+kz$
  9. The Mahler measure of $(x+1)(y+1)+kz$
  10. The probability density function of the areal Mahler measure of $(x+1)(y+1)+kz$
  11. The Proof of Theorem \ref{['thm:Qk']}
  12. Further understanding of the main results
  13. Volumes of Deninger cycles
  14. Modular interpretation of $c_0(k)$

Key Result

Theorem 1.1

If $0\leq k \leq 2$, we have If $k\geq 2$, then

Figures (2)

  • Figure 3.1: The graph of $Z_{\mathbb{D}}(s, k + x + y)$
  • Figure 3.2: The graph of $Z_{\mathbb{D}}(s, 1 + x + y)$ for $-20 \leq \mathop{\mathrm{Re}}(s) \leq 10$ and $|\mathop{\mathrm{Im}}(s)| \leq 30$

Theorems & Definitions (72)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • ...and 62 more