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The Mahler measure of a family of polynomials with arbitrarily many variables

Siva Sankar Nair

TL;DR

The paper derives an exact formula for the Mahler measure of the family $Q_n$ of polynomials in arbitrarily many variables by transforming the Mahler integral into iterated univariate integrals and then expressing the resulting polylogarithm values at sixth roots of unity in terms of zeta-values and the Dirichlet $L$-function $L( ext{χ}_{-3},s)$. Central to the method are hyperlogarithms and multiple polylogarithms, plus two complementary evaluation schemes for the one-variable integrals: a recursive polynomial framework and an explicit non-recursive approach via parameter derivatives. The final result expresses $m(Q_{2n})$ and $m(Q_{2n+1})$ as $Q$-linear combinations of $oldsymbol{eta}$-values, namely $oldsymbol{ abla}$-type combinations of $oldsymbol{ ext{ζ}}(2h+1)$ and $L( ext{χ}_{-3},2h+2)$, with coefficients $a_{r,s}, b_{r,s}, c_{r,s}, d_{r,s}$ constrained to lie in $Q$ or $Q( frac{}{} oot 3)$. The work also provides an alternate, explicit polynomial route to these coefficients and discusses the potential for extending the framework to other roots of unity and conductors. Overall, the results illuminate how multivariable Mahler measures can encode special values of $L$-functions beyond the conductor-4 case, via polylogarithms at sixth roots of unity, and suggest fruitful directions for generalizations and closed-form coefficient formulas.

Abstract

We present an exact formula for the Mahler measure of an infinite family of polynomials with arbitrarily many variables. The formula is obtained by manipulating the integral defining the Mahler measure using certain transformations, followed by an iterative process that reduces this computation to the evaluation of certain polylogarithm functions at sixth roots of unity. This yields values of the Riemann zeta function and the Dirichlet $L$-function associated to the character of conductor 3.

The Mahler measure of a family of polynomials with arbitrarily many variables

TL;DR

The paper derives an exact formula for the Mahler measure of the family of polynomials in arbitrarily many variables by transforming the Mahler integral into iterated univariate integrals and then expressing the resulting polylogarithm values at sixth roots of unity in terms of zeta-values and the Dirichlet -function . Central to the method are hyperlogarithms and multiple polylogarithms, plus two complementary evaluation schemes for the one-variable integrals: a recursive polynomial framework and an explicit non-recursive approach via parameter derivatives. The final result expresses and as -linear combinations of -values, namely -type combinations of and , with coefficients constrained to lie in or . The work also provides an alternate, explicit polynomial route to these coefficients and discusses the potential for extending the framework to other roots of unity and conductors. Overall, the results illuminate how multivariable Mahler measures can encode special values of -functions beyond the conductor-4 case, via polylogarithms at sixth roots of unity, and suggest fruitful directions for generalizations and closed-form coefficient formulas.

Abstract

We present an exact formula for the Mahler measure of an infinite family of polynomials with arbitrarily many variables. The formula is obtained by manipulating the integral defining the Mahler measure using certain transformations, followed by an iterative process that reduces this computation to the evaluation of certain polylogarithm functions at sixth roots of unity. This yields values of the Riemann zeta function and the Dirichlet -function associated to the character of conductor 3.
Paper Structure (21 sections, 4 theorems, 258 equations, 2 figures, 1 table)

This paper contains 21 sections, 4 theorems, 258 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hyperlogarithms and multiple polylogarithms
  3. Some general integrals
  4. The integral $f_1(k)$
  5. The integral $f_2(k)$
  6. The integral $g_1(k)$
  7. The integral $g_2(k)$
  8. Dependence on the sign of $a$ and $b$
  9. An alternate method
  10. The integral $f_1(k)$
  11. The integral $f_2(k)$
  12. The integral $g_1(k)$
  13. The integral $g_2(k)$
  14. Summary
  15. Computing the Mahler measure
  16. ...and 6 more sections

Key Result

Theorem 1

(nvarLL) For $n \geq 1$, where For $n \geq 0$, where

Figures (2)

  • Figure 1: The keyhole contour $C$ for $f_j(k)$ - the black dots denote the poles for $f_1(k)$, while the red dots denote the poles for $f_2(k)$.
  • Figure 2: The contour $C'$ for $g_j(k)$ - the black dots denote the poles for $g_1(k)$, while the red dots denote the poles for $g_2(k)$. Note that $z=a$ is outside the contour.

Theorems & Definitions (8)

  • Theorem
  • Theorem 1
  • Proposition 2
  • Remark 3
  • proof : Proof of Theorem \ref{['thm:main']}
  • Proposition 4
  • Remark 5
  • proof : Proof of Proposition