Mahler measure explained

In mathematics, the Mahler measure

M(p) of a polynomial p(z) with complex coefficients is defined as

$$M(p)\; =\; |a|\backslash prod\_$$

\ge 1

|\alpha_i| = |a| \prod_^n \max\,where p(z) factorizes over the complex numbers C as$$p(z)\; =\; a(z-\backslash alpha\_1)(z-\backslash alpha\_2)\backslash cdots(z-\backslash alpha\_n).$$

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 ):$$M(p)\; =\; \backslash exp\backslash left(\backslash int\_^\; \backslash ln(|p(e^)|)\backslash ,\; d\backslash theta\; \backslash right).$$

By extension, the Mahler measure of an algebraic number

\alpha is defined as the Mahler measure of the minimal polynomial of \alpha over 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 ⋅ q) = M(p) ⋅ M(q).

• $M(p)\; =\; \backslash lim\_\; \backslash |p\backslash |\_$ where $\backslash ,\; \backslash |p\backslash |\_\backslash tau\; =\backslash left(\backslash int\_0^\; |p(e^)|^\backslash tau\; d\backslash theta\; \backslash right)^$ 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 C [x _{1,\ldots,x} _{n]} is defined similarly by the formula^[2]

$$M(p)\; =\; \backslash exp\backslash left(\backslash int\_0^\; \backslash int\_0^\; \backslash cdots\; \backslash int\_0^\; \backslash log\; \backslash Bigl(\backslash bigl\; |p(e^,\; e^,\; \backslash ldots,\; e^)\; \backslash bigr|\; \backslash Bigr)\; \backslash ,\; d\backslash theta\_1\backslash ,\; d\backslash theta\_2\backslash cdots\; d\backslash theta\_n\; \backslash right).$$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 valuesof zeta-functions and

L -functions. For example, in 1981, Smyth^[3] proved the formulas$$m(1+x+y)=\backslash fracL(\backslash chi\_,2)$$where

L(\chi _{-3} ,s) is a Dirichlet L-function, and$$m(1+x+y+z)=\backslash frac\backslash zeta(3),$$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

Z denote the integers and define Z ^{N} _{+= \{r = (r} _{1,...,r} ^{N:r} ^{j\ge 0 for 1 \le} j\le N\} . If Q(z _{1,...,z} _{N)} is a polynomial in N variables and r = (r _{1,...,r} ^{N} ^{+} define the polynomial Q _{r(z)} of one variable by

$$Q\_r(z):=Q(z^,\backslash dots,z^)$$

and define

q(r) by$$q(r)\; :=\; \backslash min\; \backslash left\backslash$$

where

H(s) =max \{|s _{j|:1\le} j\le N\} .

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.

Define an extended cyclotomic polynomial to be a polynomial of the form$$\backslash Psi(z)=z\_1^\; \backslash dots\; z\_n^\backslash Phi\_m(z\_1^\backslash dots\; z\_n^),$$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 ^{c _{1}} _{1} ... z ^{c _{n}} _{n} and extended cyclotomic polynomials.

This led Boyd to consider the set of values$$L\_n:=\backslash bigl\backslash ,$$and the union $\_\backslash infty\; =\; \backslash bigcup^\backslash infty\_L\_n$. He made the far-reaching conjecture that the set of

{L} _{infty} is a closed subset of 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$$L\_1\backslash subsetneqq\; L\_2\backslash subsetneqq\; L\_3\backslash subsetneqq\backslash \; \backslash cdots\; .$$

Mahler measure and entropy

An action

\alpha _{M} of Z ^{n} by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module N over the ring R = Z [z ^{\pm 1} _{1} , ... ,z ^{\pm 1} _{n} ] .^[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,...,z} _{n)\inZ[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 , inf ty] or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus 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

References

• Book: Borwein , Peter . Peter Borwein . Computational Excursions in Analysis and Number Theory . CMS Books in Mathematics . 10 . . 2002 . 3, 15 . 978-0-387-95444-8 . 1020.12001 .
• David . Boyd . David William Boyd . Speculations concerning the range of Mahler's measure . 453–469 . . 24 . 4 . 1981a . 10.4153/cmb-1981-069-5 . free.
• David . Boyd . David William Boyd . Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables . 116–121 . . 13 . 1981b . 10.1016/0022-314x(81)90033-0. free .
• Book: Boyd , David . David William Boyd . Mahler's measure and invariants of hyperbolic manifolds . Number theory for the Millenium . M. A. . Bennett . A. K. Peters . 127–143 . 2002a .
• David . Boyd . David William Boyd . Mahler's measure, hyperbolic manifolds and the dilogarithm . 3–4, 26–28 . Canadian Mathematical Society Notes . 34 . 2 . 2002b .
• David . Boyd . David William Boyd . Fernando . Rodriguez Villegas . Mahler's measure and the dilogarithm, part 1 . . 54 . 3 . 468–492 . 2002 . 10.4153/cjm-2002-016-9 . free. 10069657 .
• Book: Brunault . François . Zudilin. Wadim. Many variations of Mahler measures : a lasting symphony . Cambridge University Press . Cambridge, United Kingdom New York, NY . 2020 . 978-1-108-79445-9 . 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
• .
• J.L. . Jensen . Sur un nouvel et important théorème de la théorie des fonctions . . 22 . 359–364 . 1899 . 30.0364.02 . 10.1007/BF02417878 . free .
• Book: Knuth , Donald E. . Donald E. Knuth . 4.6.2 Factorization of Polynomials . Seminumerical Algorithms . . 2 . 3rd . Addison-Wesley . 1997 . 439–461, 678–691 . 978-0-201-89684-8 .
• Wayne M. . Lawton . A problem of Boyd concerning geometric means of polynomials . . 16 . 3 . 356–362 . 1983 . 0516.12018 . 10.1016/0022-314X(83)90063-X . free .
• Michael J. . Mossinghoff . Polynomials with Small Mahler Measure . . 67 . 224 . 1697–1706 . 1998 . 0918.11056 . 10.1090/S0025-5718-98-01006-0 . free .
• Book: Schinzel , Andrzej . Andrzej Schinzel . Polynomials with special regard to reducibility . Encyclopedia of Mathematics and Its Applications . 77 . . 2000 . 978-0-521-66225-3 . 0956.12001 . registration .
• Book: Smyth , Chris . The Mahler measure of algebraic numbers: a survey . 322–349 . James . McKee . Smyth . Chris . Number Theory and Polynomials . London Mathematical Society Lecture Note Series . 352 . . 2008 . 978-0-521-71467-9 . 1334.11081 .

External links

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

Notes and References

1. Although this is not a true norm for values of \tau < 1 .
2. .
3. .
4. .
5. .
6. .
7. Kitchens. Bruce. Schmidt. Klaus. 1989. Automorphisms of compact groups . Ergodic Theory and Dynamical Systems . 9. 4. 691–735. 10.1017/S0143385700005290. free.
8. Lind. Douglas. Schmidt. Klaus . Ward. Tom. 1990. Mahler measure and entropy for commuting automorphisms of compact groups. Inventiones Mathematicae. 101. 593–629. 10.1007/BF01231517. free.
9. Lind. Douglas. 1977. The structure of skew products with ergodic group automorphisms. Israel Journal of Mathematics. 28. 3. 205–248. 10.1007/BF02759810. 120160631 .