Mahler measure

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

$${\displaystyle M(p)=|a|\prod _{|\alpha _{i}|\geq 1}|\alpha _{i}|=|a|\prod _{i=1}^{n}\max\{1,|\alpha _{i}|\},}$$

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

$${\displaystyle p(z)=a(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n}).}$$

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 ${\displaystyle |p(z)|}$ for ${\displaystyle z}$ on the unit circle (i.e., ${\displaystyle |z|=1}$):

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

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

The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.

Properties

• The Mahler measure is multiplicative: ${\displaystyle \forall p,q,\,\,M(p\cdot q)=M(p)\cdot M(q).}$
• ${\textstyle M(p)=\lim _{\tau \to 0}\|p\|_{\tau }}$ where ${\textstyle \,\|p\|_{\tau }=\left(\int _{0}^{1}|p(e^{2\pi i\theta })|^{\tau }d\theta \right)^{1/\tau }}$ is the ${\displaystyle L_{\tau }}$ norm of ${\displaystyle p}$. ^[1]
• Kronecker's Theorem: If ${\displaystyle p}$ is an irreducible monic integer polynomial with ${\displaystyle M(p)=1}$, then either ${\displaystyle p(z)=z,}$ or ${\displaystyle p}$ is a cyclotomic polynomial.
• (Lehmer's conjecture) There is a constant ${\displaystyle \mu >1}$ such that if ${\displaystyle p}$ is an irreducible integer polynomial, then either ${\displaystyle M(p)=1}$ or ${\displaystyle M(p)>\mu }$.
• The Mahler measure of a monic integer polynomial is a Perron number.

Higher-dimensional Mahler measure

The Mahler measure ${\displaystyle M(p)}$ of a multi-variable polynomial ${\displaystyle p(x_{1},\ldots ,x_{n})\in \mathbb {C} [x_{1},\ldots ,x_{n}]}$ is defined similarly by the formula ^[2]

$${\displaystyle M(p)=\exp \left(\int _{0}^{1}\int _{0}^{1}\cdots \int _{0}^{1}\log {\Bigl (}{\bigl |}p(e^{2\pi i\theta _{1}},e^{2\pi i\theta _{2}},\ldots ,e^{2\pi i\theta _{n}}){\bigr |}{\Bigr )}\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n}\right).}$$

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 ${\displaystyle L}$-functions. For example, in 1981, Smyth ^[3] proved the formulas

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

where ${\displaystyle L(\chi _{-3},s)}$ is a Dirichlet L-function, and

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

where ${\displaystyle \zeta }$ is the Riemann zeta function. Here ${\displaystyle m(P)=\log M(P)}$ 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 ${\displaystyle p}$ vanishes on the torus ${\displaystyle (S^{1})^{n}}$, then the convergence of the integral defining ${\displaystyle M(p)}$ is not obvious, but it is known that ${\displaystyle M(p)}$ 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 ${\displaystyle \mathbb {Z} }$ denote the integers and define ${\displaystyle \mathbb {Z} _{+}^{N}=\{r=(r_{1},\dots ,r_{N})\in \mathbb {Z} ^{N}:r_{j}\geq 0\ {\text{for}}\ 1\leq j\leq N\}}$ . If ${\displaystyle Q(z_{1},\dots ,z_{N})}$ is a polynomial in ${\displaystyle N}$ variables and ${\displaystyle r=(r_{1},\dots ,r_{N})\in \mathbb {Z} _{+}^{N}}$ define the polynomial ${\displaystyle Q_{r}(z)}$ of one variable by

$${\displaystyle Q_{r}(z):=Q(z^{r_{1}},\dots ,z^{r_{N}})}$$

and define ${\displaystyle q(r)}$ by

$${\displaystyle q(r):=\min \left\{H(s):s=(s_{1},\dots ,s_{N})\in \mathbb {Z} ^{N},s\neq (0,\dots ,0)~{\text{and}}~\sum _{j=1}^{N}s_{j}r_{j}=0\right\}}$$

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

Theorem (Lawton) — Let ${\displaystyle Q(z_{1},\dots ,z_{N})}$ be a polynomial in N variables with complex coefficients. Then the following limit is valid (even if the condition that ${\displaystyle r_{i}\geq 0}$ is relaxed):

$${\displaystyle \lim _{q(r)\to \infty }M(Q_{r})=M(Q)}$$

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

$${\displaystyle \Psi (z)=z_{1}^{b_{1}}\dots z_{n}^{b_{n}}\Phi _{m}(z_{1}^{v_{1}}\dots z_{n}^{v_{n}}),}$$

where ${\displaystyle \Phi _{m}(z)}$ is the m-th cyclotomic polynomial, the ${\displaystyle v_{i}}$ are integers, and the ${\displaystyle b_{i}=\max(0,-v_{i}\deg \Phi _{m})}$ are chosen minimally so that ${\displaystyle \Psi (z)}$ is a polynomial in the ${\displaystyle z_{i}}$. Let ${\displaystyle K_{n}}$ be the set of polynomials that are products of monomials ${\displaystyle \pm z_{1}^{c_{1}}\dots z_{n}^{c_{n}}}$ and extended cyclotomic polynomials.

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

This led Boyd to consider the set of values

$${\displaystyle L_{n}:={\bigl \{}m(P(z_{1},\dots ,z_{n})):P\in \mathbb {Z} [z_{1},\dots ,z_{n}]{\bigr \}},}$$

and the union ${\textstyle {L}_{\infty }=\bigcup _{n=1}^{\infty }L_{n}}$. He made the far-reaching conjecture ^[5] that the set of ${\displaystyle {L}_{\infty }}$ is a closed subset of ${\displaystyle \mathbb {R} }$. 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 ${\displaystyle L_{1}\subsetneqq L_{2}}$ , Boyd further conjectures that

$${\displaystyle L_{1}\subsetneqq L_{2}\subsetneqq L_{3}\subsetneqq \ \cdots .}$$

Mahler measure and entropy

An action ${\displaystyle \alpha _{M}}$ of ${\displaystyle \mathbb {Z} ^{n}}$ by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module ${\displaystyle N}$ over the ring ${\displaystyle R=\mathbb {Z} [z_{1}^{\pm 1},\dots ,z_{n}^{\pm 1}]}$. ^[7] The topological entropy (which is equal to the measure-theoretic entropy) of this action, ${\displaystyle h(\alpha _{N})}$, is given by a Mahler measure (or is infinite). ^[8] In the case of a cyclic module ${\displaystyle M=R/\langle F\rangle }$ for a non-zero polynomial ${\displaystyle F(z_{1},\dots ,z_{n})\in \mathbb {Z} [z_{1},\ldots ,z_{n}]}$ the formula proved by Lind, Schmidt, and Ward gives ${\displaystyle h(\alpha _{N})=\log M(F)}$, the logarithmic Mahler measure of ${\displaystyle F}$. In the general case, the entropy of the action is expressed as a sum of logarithmic Mahler measures over the generators of the principal associated prime ideals of the module. As pointed out earlier by Lind in the case ${\displaystyle n=1}$ of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of ${\displaystyle [0,\infty ]}$ or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus ${\displaystyle \mathbb {T} ^{\infty }}$ either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem. ^[9]

See also

• Bombieri norm
• Height of a polynomial

Notes

1. Although this is not a true norm for values of ${\displaystyle \tau <1}$.
2. Schinzel 2000 , p. 224.
3. Smyth 2008.
4. Lawton 1983.
5. Boyd 1981a.
6. Boyd 1981b.
7. Kitchens, Bruce; Schmidt, Klaus (1989). "Automorphisms of compact groups". Ergodic Theory and Dynamical Systems. 9 (4): 691–735. doi: 10.1017/S0143385700005290 .
8. Lind, Douglas; Schmidt, Klaus; Ward, Tom (1990). "Mahler measure and entropy for commuting automorphisms of compact groups". Inventiones Mathematicae. 101: 593–629. doi: 10.1007/BF01231517 .
9. Lind, Douglas (1977). "The structure of skew products with ergodic group automorphisms". Israel Journal of Mathematics . 28 (3): 205–248. doi:10.1007/BF02759810. S2CID   120160631.

References

• Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Vol. 10. Springer. pp. 3, 15. ISBN   978-0-387-95444-8. Zbl   1020.12001.

• Boyd, David (1981a). "Speculations concerning the range of Mahler's measure". Canadian Mathematical Bulletin . 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. (ed.). 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, Fernando (2002). "Mahler's measure and the dilogarithm, part 1". Canadian Journal of Mathematics . 54 (3): 468–492. doi: 10.4153/cjm-2002-016-9 . S2CID   10069657.
• Brunault, François; Zudilin, Wadim (2020). Many variations of Mahler measures : a lasting symphony. Cambridge, United Kingdom New York, NY: Cambridge University Press. ISBN   978-1-108-79445-9. OCLC   1155888228.

• Everest, Graham and Ward, Thomas (1999). "Heights of polynomials and entropy in algebraic dynamics". Springer-Verlag London, Ltd., London. xii+211 pp. ISBN: 1-85233-125-9

• "Mahler measure", Encyclopedia of Mathematics , EMS Press, 2001 [1994].
• 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. Vol. 2 (3rd ed.). Addison-Wesley. pp. 439–461, 678–691. ISBN   978-0-201-89684-8.

• Lawton, Wayne M. (1983). "A problem of Boyd concerning geometric means of polynomials". Journal of Number Theory . 16 (3): 356–362. doi: 10.1016/0022-314X(83)90063-X . Zbl   0516.12018.

• Mossinghoff, Michael 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. Vol. 77. Cambridge University Press. ISBN   978-0-521-66225-3. Zbl   0956.12001.

• Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris (eds.). Number Theory and Polynomials. London Mathematical Society Lecture Note Series. Vol. 352. Cambridge University Press. pp. 322–349. ISBN   978-0-521-71467-9. Zbl   1334.11081.

External links

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