Mahler measure

In mathematics, the Mahler measure $M(p)$ of a polynomial $p(z)$ with complex coefficients is

$M(p) = |a|\prod_{|\alpha_i| \ge 1} |\alpha_i| = |a| \prod_{i=1}^n \max\{1,|\alpha_i|\},$

where $p(z)$ factorizes over the complex numbers $\mathbb{C}$ as

$p(z) = a(z-\alpha_1)(z-\alpha_2)\cdots(z-\alpha_n).$

It can be shown using Jensen's formula that also

$M(p) = \exp\left( \frac{1}{2\pi} \int_{0}^{2\pi} \ln(|p(e^{i\theta})|)\, d\theta \right),$

which is the geometric mean of $|p(z)|$ for $z$ on the unit circle $|z| = 1.$

The Mahler measure of an algebraic number $\alpha$ is defined as the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb{Q}$. In particular, if $\alpha$ is a Pisot number or a Salem number then its Mahler measure is simply $\alpha$.

The Mahler measure is named after Kurt Mahler. It is in fact a kind of Height function.

Properties

• The Mahler measure is multiplicative, i.e. $M(p\,q) = M(p) \cdot M(q).$
• Also $M(p) = \lim_{\tau \rightarrow 0} \|p\|_{\tau},$ where

$\|p\|_\tau =\left( \frac{1}{2\pi} \int_0^{2\pi} |p(e^{i\theta})|^\tau \, d\theta \right)^{1/\tau} \,$

is the $L_\tau$ norm of $p$ (although this is not a true norm for values of $\tau < 1$).

• (Kronecker's Theorem) If $p$ is an irreducible monic integer polynomial with $M(p) = 1$, then either $p(z) = z,$ or $p$ is a cyclotomic polynomial.
• Lehmer's conjecture asserts that there is a constant $\mu>1$ such that if $p$ is an irreducible integer polynomial, then either $M(p)=1$ or $M(p)>\mu$.
• The Mahler measure of a monic integer polynomial is a Perron number.

Higher-dimensional Mahler measure

The Mahler measure $M(p)$ of a multi-variable polynomial $p(x_1,\ldots,x_n) \in \mathbb{C}[x_1,\ldots,x_n]$ is defined similarly by the formula

$M(p) = \exp\left( \frac{1}{(2\pi)^n} \int_0^{2\pi} \int_0^{2\pi} \cdots \int_0^{2\pi} \log \Bigl( \bigl |p(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n}) \bigr| \Bigr) \, d\theta_1\, d\theta_2\cdots d\theta_n \right).$

They inherit the above three properties of the Mahler measure for a one-variable polynomial.

The multi-variable Mahler measure has been shown, in some cases, to be related to special values of zeta-functions and $L$-functions. For example, In 1981 Chris Smyth proved the formulas

$m(1+x+y)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2)$

where $L(\chi_{-3},s)$ is the Dirichlet L-function, and

$m(1+x+y+z)=\frac{7}{2\pi^2}\zeta(3)$ ,

where $\zeta$ is the Riemann zeta function. Here $m(P)=\log{M(P)}$ is called the logarithmic Mahler measure.

Some results by Lawton and Boyd

From the definition the Mahler measure is viewed as the integrated values of polynomials over the torus (also see Lehmer's conjecture). If $p$ vanishes on the torus $(S^1)^n$, then the convergence of the integral defining $M(p)$ is not obvious, but it is known that $M(p)$ does converge and is equal to a limit of one-variable Mahler measures, which had been conjectured by D. Boyd.

This is formulated as follows: Let $\mathbb{Z}$ denote the integers and define $\mathbb{Z}^N_+=\{r=(r_1,\dots,r_N)\in\mathbb{Z}^N:r_j\ge0\ \text{for}\ 1\le j\le N\}$ . If $Q(z_1,\dots,z_N)$ is a polynomial in $N$ variables and $r=(r_1,\dots,r_N)\in\mathbb{Z}^N_+$ define the polynomial $Q_r(z)$ of one variable by

$Q_r(z):=Q(z^{r_1},\dots,z^{r_N})$

and define $q(r)$ by

$q(r):=\text{min}\{H(s):s=(s_1,\dots,s_N)\in\mathbb{Z}^N,s\ne(0,\dots,0)\ \text{and}\ \sum^N_{j=1}s_jr_j=0\}$

where $H(s)=\text{max}\{|s_j|:1\le j\le N\}$ .

Theorem (Lawton) : Let $Q(z_1,\dots,z_N)$ be a polynomial in N variables with complex coefficients. Then the following limit is valid (even if the condition that $r_i\ge0$ is relaxed):

$\lim_{q(r)\rightarrow\infty}M(Q_r)=M(Q)$

Boyd's proposal

D. Boyd provided more general statements than the above theorem. He pointed out that the classical Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk, can be regarded as characterizing those polynomials of one variable whose measure is exactly 1, and that this result extends to polynomials in several variables.

Define an extended cyclotomic polynomial to be a polynomial of the form

$\Psi(z)=z_1^{b_1} \dots z_n^{b_n}\Phi_m(z_1^{v_1}\dots z_n^{v_n}),$

where $\Phi_m(z)$ is the m-th cyclotomic polynomial, the $v_i$ are integers, and the $b_i=\max(0,-v_i\deg\Phi_m)$ are chosen minimally so that $\Psi(z)$ is a polynomial in the $z_i$. Let $K_n$ be the set of polynomials that are products of monomials $\pm z_1^{c_1}\dots z_n^{c_n}$ and extended cyclotomic polynomials.

Theorem (Boyd) : Let $F(z_1,\dots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n]$ be a polynomial with integer coefficients. Then $M(F)=1$ if and only if $F$ is an element of $K_n$.

This led Boyd to consider the set of values

$L_n:=\bigl\{m(P(z_1,\dots,z_n)):P\in\mathbb{Z}[z_1,\dots,z_n]\bigr\},$

and the union ${L}_\infty=\bigcup^\infty_{n=1}L_n$. He made the far-reaching conjecture that the set of ${L}_\infty$ is a closed subset of $\mathbb R$. An immediate consequence of this conjecture would be the truth of Lehmer's conjecture, albeit without an explicit lower bound. As Smyth's result suggests that $L_1\subsetneqq L_2$ , Boyd further conjectures that

$L_1\subsetneqq L_2\subsetneqq L_3\subsetneqq\ \cdots\,.$

See also

• Bombieri norm
• Height of a polynomial

References

• Hazewinkel, Michiel, ed. (2001), Mahler measure, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 [1]
• David Boyd and F. Rodriguez Villegas: Mahler's measure and the dilogarithm, part 1, Canadian J. Math., vol, 54, 2002, pp. 468–492

External links

• Mahler Measure on MathWorld
• Jensen's Formula on MathWorld