Mahler measure

{{Use American English|date = March 2019}} {{Short description|Measure of polynomial height}} In mathematics, the '''Mahler measure''' <math>M(p)</math> '''of a polynomial''' <math>p(z) </math> with complex coefficients is defined as

<math display="block">M(p) = |a|\prod_{|\alpha_i| \ge 1} |\alpha_i| = |a| \prod_{i=1}^n \max\{1,|\alpha_i|\},</math> where <math>p(z) </math> factorizes over the complex numbers <math>\mathbb{C}</math> as <math display="block">p(z) = a(z-\alpha_1)(z-\alpha_2)\cdots(z-\alpha_n).</math>

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 <math>|p(z)| </math> for <math>z</math> on the unit circle (i.e., <math>|z| = 1</math>): <math display="block">M(p) = \exp\left(\int_{0}^{1} \ln(|p(e^{2\pi i\theta})|)\, d\theta \right).</math>

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

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

== Properties ==

* The Mahler measure is multiplicative: <math>\forall p, q, \,\, M(p \cdot q) = M(p) \cdot M(q).</math> * <math display="inline">M(p) = \lim_{\tau \to 0} \|p\|_{\tau}</math> where <math display="inline">\, \|p\|_\tau =\left(\int_0^{1} |p(e^{2\pi i\theta})|^\tau d\theta \right)^{1/\tau} </math> is the <math>L_\tau</math> norm of <math>p</math>.<ref>Although this is not a true norm for values of <math>\tau < 1</math>.</ref> * '''Kronecker's Theorem''': If <math>p</math> is an irreducible monic integer polynomial with <math>M(p) = 1</math>, then either <math>p(z) = z,</math> or <math>p</math> is a cyclotomic polynomial. * ''(Lehmer's conjecture)'' There is a constant <math>\mu>1</math> such that if <math>p</math> is an irreducible integer polynomial, then either <math>M(p)=1</math> or <math>M(p)>\mu</math>. * The Mahler measure of a monic integer polynomial is a Perron number.

== Higher-dimensional Mahler measure ==

The Mahler measure <math>M(p)</math> of a multi-variable polynomial <math>p(x_1,\ldots,x_n) \in \mathbb{C}[x_1,\ldots,x_n]</math> is defined similarly by the formula<ref name=Sch224>{{harvnb|Schinzel|2000|page=224}}.</ref>

<math display="block">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).</math> 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 <math>L</math>-functions. For example, in 1981, Smyth<ref>{{harvnb|Smyth|2008}}.</ref> proved the formulas <math display="block"> m(1+x+y)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2)</math> where <math>L(\chi_{-3},s)</math> is a Dirichlet L-function, and <math display="block"> m(1+x+y+z)=\frac{7}{2\pi^2}\zeta(3),</math> where <math>\zeta</math> is the Riemann zeta function. Here <math> m(P)=\log M(P)</math> 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 <math>p</math> vanishes on the torus <math>(S^1)^n</math>, then the convergence of the integral defining <math>M(p)</math> is not obvious, but it is known that <math>M(p)</math> does converge and is equal to a limit of one-variable Mahler measures,<ref name=Law83>{{harvnb|Lawton|1983}}.</ref> which had been conjectured by Boyd.<ref name=Boyd1981a>{{harvnb|Boyd|1981a}}.</ref><ref name=Boyd1981b>{{harvnb|Boyd|1981b}}.</ref>

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

and define <math>q(r)</math> by <math display="block">q(r) := \min \left\{H(s):s=(s_1,\dots,s_N)\in\mathbb{Z}^N, s\ne(0,\dots,0)~\text{and}~\sum^N_{j=1} s_j r_j = 0 \right\}</math>

where <math>H(s)=\max\{|s_j|:1\le j\le N\}</math>.

{{math theorem | name = Theorem (Lawton) | math_statement = Let <math>Q(z_1,\dots,z_N)</math> be a polynomial in ''N'' variables with complex coefficients. Then the following limit is valid (even if the condition that <math>r_i \geq 0</math> is relaxed): <math display="block"> \lim_{q(r)\to\infty}M(Q_r)=M(Q)</math>}}

===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.<ref name=Boyd1981b />

Define an ''extended cyclotomic polynomial'' to be a polynomial of the form <math display="block">\Psi(z)=z_1^{b_1} \dots z_n^{b_n}\Phi_m(z_1^{v_1}\dots z_n^{v_n}),</math> where <math>\Phi_m(z)</math> is the ''m''-th cyclotomic polynomial, the <math>v_i</math> are integers, and the <math>b_i = \max(0, -v_i\deg\Phi_m)</math> are chosen minimally so that <math>\Psi(z)</math> is a polynomial in the <math>z_i</math>. Let <math>K_n</math> be the set of polynomials that are products of monomials <math>\pm z_1^{c_1}\dots z_n^{c_n}</math> and extended cyclotomic polynomials.

{{math theorem | name = Theorem (Boyd) | math_statement = Let <math> F(z_1,\dots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n]</math> be a polynomial with integer coefficients. Then <math>M(F)=1</math> if and only if <math>F</math> is an element of <math>K_n</math>.}}

This led Boyd to consider the set of values <math display="block">L_n:=\bigl\{m(P(z_1,\dots,z_n)):P\in\mathbb{Z}[z_1,\dots,z_n]\bigr\},</math> and the union <math display="inline">{L}_\infty = \bigcup^\infty_{n=1}L_n</math>. He made the far-reaching conjecture<ref name=Boyd1981a /> that the set of <math>{L}_\infty</math> is a closed subset of <math>\mathbb R</math>. 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 <math>L_1\subsetneqq L_2</math> , Boyd further conjectures that <math display="block">L_1\subsetneqq L_2\subsetneqq L_3\subsetneqq\ \cdots .</math>

=== Mahler measure and entropy === An action <math>\alpha_M</math> of <math>\mathbb{Z}^n</math> by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module <math>N</math> over the ring <math>R=\mathbb{Z}[z_1^{\pm1},\dots,z_n^{\pm1}]</math>.<ref>{{Cite journal|last1=Kitchens| first1=Bruce|last2=Schmidt|first2=Klaus| date=1989| title=Automorphisms of compact groups |journal=Ergodic Theory and Dynamical Systems |volume=9|issue=4|pages=691–735|doi=10.1017/S0143385700005290|doi-access=free}}</ref> The topological entropy (which is equal to the measure-theoretic entropy) of this action, <math>h(\alpha_N)</math>, is given by a Mahler measure (or is infinite).<ref>{{Cite journal|last1=Lind|first1=Douglas|last2=Schmidt|first2=Klaus |last3=Ward|first3=Tom|date=1990|title=Mahler measure and entropy for commuting automorphisms of compact groups|journal=Inventiones Mathematicae| volume=101| pages=593–629|doi=10.1007/BF01231517|doi-access=free|bibcode=1990InMat.101..593L }}</ref> In the case of a cyclic module <math>M=R/\langle F\rangle</math> for a non-zero polynomial <math> F(z_1,\dots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n]</math> the formula proved by Lind, Schmidt, and Ward gives <math>h(\alpha_N)=\log M(F)</math>, the logarithmic Mahler measure of <math>F</math>. 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 <math>n=1</math> of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of <math>[0,\infty]</math> or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus <math>\mathbb{T}^{\infty}</math> either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem.<ref>{{Cite journal|last=Lind|first=Douglas| date=1977|title=The structure of skew products with ergodic group automorphisms| journal=Israel Journal of Mathematics|volume=28|issue=3|pages=205–248|doi=10.1007/BF02759810|s2cid=120160631 |doi-access=}}</ref>

== See also ==

*Bombieri norm *Height of a polynomial

== Notes ==

== References ==

*{{cite book | last = Borwein | first = Peter | author-link = Peter Borwein | title = Computational Excursions in Analysis and Number Theory | series = CMS Books in Mathematics | volume = 10 | publisher = Springer | year = 2002 | pages = 3, 15 | isbn = 978-0-387-95444-8 | zbl = 1020.12001 }}

*{{cite journal | first = David | last = Boyd | author-link = David William Boyd | title = Speculations concerning the range of Mahler's measure | pages = 453–469 | journal = Canadian Mathematical Bulletin | volume = 24 | issue = 4 | year = 1981a | doi=10.4153/cmb-1981-069-5 | doi-access=free }}

*{{cite journal | first = David | last = Boyd | author-link = David William Boyd | title = Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables | pages = 116–121 | journal = Journal of Number Theory | volume = 13 | year = 1981b | doi=10.1016/0022-314x(81)90033-0 | doi-access = free }}

*{{cite book | first = David | last = Boyd | author-link = David William Boyd | chapter = Mahler's measure and invariants of hyperbolic manifolds | title= Number theory for the Millenium | editor-first = M. A. | editor-last = Bennett | publisher = A. K. Peters | pages = 127–143 | year = 2002a }}

*{{cite journal | first = David | last = Boyd | author-link = David William Boyd | title = Mahler's measure, hyperbolic manifolds and the dilogarithm | pages = 3–4, 26–28 | journal = Canadian Mathematical Society Notes | volume = 34 | number = 2 | year = 2002b }}

*{{cite journal | first1 = David | last1 = Boyd | author1-link = David William Boyd | first2 = Fernando | last2 = Rodriguez Villegas | title = Mahler's measure and the dilogarithm, part 1 | journal = Canadian Journal of Mathematics | volume = 54 | issue = 3 | pages = 468–492 | year = 2002 | doi=10.4153/cjm-2002-016-9 | doi-access=free | s2cid = 10069657 }} * {{cite book | last1=Brunault | first1=François | last2=Zudilin | first2=Wadim | title=Many variations of Mahler measures : a lasting symphony | publisher=Cambridge University Press | publication-place=Cambridge, United Kingdom New York, NY | year=2020 | isbn=978-1-108-79445-9 | oclc=1155888228}}

*Everest, Graham and Ward, Thomas (1999). [https://link.springer.com/book/10.1007/978-1-4471-3898-3 "Heights of polynomials and entropy in algebraic dynamics"]. [https://www.springer.com/gb Springer-Verlag London], Ltd., London. xii+211 pp. ISBN: 1-85233-125-9

* {{SpringerEOM||id=Mahler_measure&oldid=36175|title=Mahler measure}}. *{{cite journal | first = J.L. | last = Jensen | title = Sur un nouvel et important théorème de la théorie des fonctions | journal = Acta Mathematica | volume = 22 | pages = 359–364 | year = 1899 | jfm = 30.0364.02 | doi = 10.1007/BF02417878 | doi-access = free }}

*{{cite book | last = Knuth | first = Donald E. | author-link = Donald E. Knuth | chapter = 4.6.2 Factorization of Polynomials | title = Seminumerical Algorithms | series = The Art of Computer Programming | volume = 2 | edition = 3rd | publisher = Addison-Wesley | year = 1997 | pages = 439–461, 678–691 | isbn = 978-0-201-89684-8 }}

*{{cite journal | first = Wayne M. | last = Lawton | title = A problem of Boyd concerning geometric means of polynomials | journal = Journal of Number Theory | volume = 16 | issue = 3 | pages = 356–362 | year = 1983 | zbl = 0516.12018 | doi = 10.1016/0022-314X(83)90063-X | doi-access = free }}

*{{cite journal | first = Michael J. | last = Mossinghoff | title = Polynomials with Small Mahler Measure | journal = Mathematics of Computation | volume = 67 | issue = 224 | pages = 1697–1706 | year = 1998 | zbl = 0918.11056 | doi = 10.1090/S0025-5718-98-01006-0 | doi-access = free }}

*{{cite book | last = Schinzel | first = Andrzej | author-link = Andrzej Schinzel | title = Polynomials with special regard to reducibility | series = Encyclopedia of Mathematics and Its Applications | volume = 77 | publisher = Cambridge University Press | year = 2000 | isbn = 978-0-521-66225-3 | zbl = 0956.12001 | url-access = registration | url = https://archive.org/details/polynomialswiths0000schi }}

*{{cite book | first = Chris | last = Smyth | chapter = The Mahler measure of algebraic numbers: a survey | pages = 322–349 | editor1-first = James | editor1-last = McKee | editor2-last = Smyth | editor2-first = Chris | title = Number Theory and Polynomials | series = London Mathematical Society Lecture Note Series | volume = 352 | publisher = Cambridge University Press | year = 2008 | isbn = 978-0-521-71467-9 | zbl = 1334.11081 }}

== External links ==

*[http://mathworld.wolfram.com/MahlerMeasure.html Mahler Measure on MathWorld] *[http://mathworld.wolfram.com/JensensFormula.html Jensen's Formula on MathWorld]

Category:Analytic number theory Category:Polynomials