Mahler measure

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

M(p)=|a|\prod _{{|\alpha _{i}|\geq 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}).

The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of $|p(z)|$ for $z$ on the unit circle (i.e., $|z|=1$ ):

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

By extension, 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 the German-born Australian mathematician Kurt Mahler.

Properties

The Mahler measure is multiplicative: $\forall p,q,\,\,M(p\cdot q)=M(p)\cdot M(q).$
$M(p)=\lim _{\tau \rightarrow 0}\|p\|_{\tau }$ 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 $p$ .^[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) 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^[2]

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).

It inherits 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, Smyth^[3] 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,^[4] which had been conjectured by Boyd.^[5]^[6]

This is formulated as follows: Let $\mathbb {Z}$ denote the integers and define ${\mathbb {Z}}_{+}^{N}=\{r=(r_{1},\dots ,r_{N})\in {\mathbb {Z}}^{N}:r_{j}\geq 0\ {\text{for}}\ 1\leq j\leq N\}$ . 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\neq (0,\dots ,0)\ {\text{and}}\ \sum _{{j=1}}^{N}s_{j}r_{j}=0\}

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

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 $r_{i}\geq 0$ is relaxed):

\lim _{{q(r)\rightarrow \infty }}M(Q_{r})=M(Q)

Boyd's proposal

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 ${L}_{\infty }=\bigcup _{{n=1}}^{\infty }L_{n}$ . He made the far-reaching conjecture^[5] 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 \,.

Notes

↑ Although this is not a true norm for values of $\tau <1$ .
↑ Schinzel 2000, p. 224.
↑ Smyth 2008.
↑ Lawton 1983.
1 2 Boyd 1981a.
1 2 Boyd 1981b.

References

Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. 10. Springer. pp. 3, 15. ISBN 0-387-95444-9. Zbl 1020.12001.

Boyd, David (1981a). "Speculations concerning the range of Mahler's measure". Canad. Math. Bull. 24 (4): 453–469. doi:10.4153/cmb-1981-069-5.

Boyd, David (1981b). "Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables". Journal of Number Theory. 13: 116–121. doi:10.1016/0022-314x(81)90033-0.

Boyd, David (2002a). "Mahler's measure and invariants of hyperbolic manifolds". In Bennett, M. A. Number theory for the Millenium. A. K. Peters. pp. 127–143.

Boyd, David (2002b). "Mahler's measure, hyperbolic manifolds and the dilogarithm". Canadian Mathematical Society Notes. 34 (2): 3–4, 26–28.

Boyd, David; Rodriguez Villegas, F. (2002). "Mahler's measure and the dilogarithm, part 1". Canadian J. Math. 54: 468–492. doi:10.4153/cjm-2002-016-9.

Hazewinkel, Michiel, ed. (2001), "Mahler measure", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 .
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 (3rd ed.). 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 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.

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