Mahler measure

Last updated October 12, 2024

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

Properties

The Mahler measure is multiplicative: $\forall p,q,\,\,M(p\cdot q)=M(p)\cdot M(q).$
${\textstyle M(p)=\lim _{\tau \to 0}\|p\|_{\tau }}$ where ${\textstyle \,\|p\|_{\tau }=\left(\int _{0}^{1}|p(e^{2\pi i\theta })|^{\tau }d\theta \right)^{1/\tau }}$ is the $L_{\tau }$ 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]

$M(p)=\exp \left(\int _{0}^{1}\int _{0}^{1}\cdots \int _{0}^{1}\log {\Bigl (}{\bigl |}p(e^{2\pi i\theta _{1}},e^{2\pi i\theta _{2}},\ldots ,e^{2\pi i\theta _{n}}){\bigr |}{\Bigr )}\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n}\right).$ 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 $L$ -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 a 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 $q(r):=\min \left\{H(s):s=(s_{1},\dots ,s_{N})\in \mathbb {Z} ^{N},s\neq (0,\dots ,0)~{\text{and}}~\sum _{j=1}^{N}s_{j}r_{j}=0\right\}$

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

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

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 $\Psi (z)=z_{1}^{b_{1}}\dots z_{n}^{b_{n}}\Phi _{m}(z_{1}^{v_{1}}\dots z_{n}^{v_{n}}),$ 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 ${\textstyle {L}_{\infty }=\bigcup _{n=1}^{\infty }L_{n}}$ . 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 $L_{1}\subsetneqq L_{2}\subsetneqq L_{3}\subsetneqq \ \cdots .$

Mahler measure and entropy

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

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.
↑ Kitchens, Bruce; Schmidt, Klaus (1989). "Automorphisms of compact groups". Ergodic Theory and Dynamical Systems. 9 (4): 691–735. doi: 10.1017/S0143385700005290 .
↑ 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 .
↑ 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.

Related Research Articles

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers $A$ is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space $under the operation of composition.$

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of $L p$ spaces.

In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory.

In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. It can be used, for example, to estimate a mixture of gaussians, or to solve the multiple linear regression problem.

In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.

In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions and they are real-analytic, symmetric under the permutation of arguments, Euclidean covariant and satisfy a property known as reflection positivity. Properties of Schwinger functions are known as Osterwalder–Schrader axioms. Schwinger functions are also referred to as Euclidean correlation functions.

In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions.

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.

Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant $such that every polynomial with integer coefficients satisfies one of the following properties:$

In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : V → W that is:

In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by Szegő.

In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value at risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class.

The Bueno-Orovio–Cherry–Fenton model, also simply called Bueno-Orovio model, is a minimal ionic model for human ventricular cells. It belongs to the category of phenomenological models, because of its characteristic of describing the electrophysiological behaviour of cardiac muscle cells without taking into account in a detailed way the underlying physiology and the specific mechanisms occurring inside the cells.

Massless free scalar bosons are a family of two-dimensional conformal field theories, whose symmetry is described by an abelian affine Lie algebra.

In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure.

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

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Although this is not a true norm for values of $\tau <1$ .

[Sch224-2] Schinzel 2000 , p. 224.

[3] Smyth 2008.

[Law83-4] Lawton 1983.

[Boyd1981a-5] 1 2 Boyd 1981a.

[Boyd1981b-6] 1 2 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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]