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 $L_\tau$ 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^[1]

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

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 $L$. For example, In 1981 Chris Smyth ^[2] 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,^[3] which had been conjectured by D. Boyd.^[4][5]

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.^[6]

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^[7] 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

1. Schinzel (2000) p.224
2. Smyth (1981)
3. Lawton (1983)
4. Boyd (1981a)
5. Boyd (1981b)
6. D. Boyd (1981b)
7. D. Boyd (1981a)

• Hazewinkel, Michiel, ed. (2001), Mahler measure, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics 10. Springer-Verlag. pp. 3, 15. ISBN 0-387-95444-9. Zbl 1020.12001.
• Jensen, J.L. (1899). "Sur un nouvel et important théorème de la théorie des fonctions". Acta Mathematica 22: 359–364. doi:10.1007/BF02417878. JFM 30.0364.02.
• Knuth, Donald E. (1997). "4.6.2 Factorization of Polynomials". Seminumerical Algorithms. The Art of Computer Programming 2 (Third ed.). Reading, Massachusetts: Addison-Wesley. pp. 439–461, 678–691. ISBN 0-201-89684-2.
• Lawton, Wayne M. (1983). "A problem of Boyd concerning geometric means of polynomials". Journal of Number Theory 16: 356–362. doi:10.1016/0022-314X(83)90063-X. Zbl 0516.12018.
• Mossinghoff, M.J. (1998). "Polynomials with Small Mahler Measure". Mathematics of Computation 67 (224): 1697–1706. doi:10.1090/S0025-5718-98-01006-0. Zbl 0918.11056.
• Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.
• Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris. Number Theory and Polynomials. London Mathematical Society Lecture Note Series 352. Cambridge University Press. pp. 322–349. ISBN 978-0-521-71467-9. Zbl 06093093.
• Boyd, David (1981a). "Speculations concerning the range of Mahler's measure". Canad. Math. Bull. 24(4): 453–469.
• Boyd, David (1981b). "Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables". Journal of Number Theory 13: 116–121.
• Boyd, David (2000). Mahler's measure and invariants of hyperbolic manifolds. Number theory for the Millenium in M. A. Bennett (ed.). A. K. Peters. pp. 127–143.
• Boyd, David (2002). "Mahler's measure, hyperbolic manifolds and the dilogarithm". Canadian Mathematical Society Notes 34 (2): 3–4, 26–28.
• 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