Mahler measure

Summary

In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as

where factorizes over the complex numbers as

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 for on the unit circle (i.e., ):

By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over . In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply .

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

PropertiesEdit

  • The Mahler measure is multiplicative:  
  •   where   is the   norm of  .[1]
  • Kronecker's Theorem: If   is an irreducible monic integer polynomial with  , then either   or   is a cyclotomic polynomial.
  • (Lehmer's conjecture) There is a constant   such that if   is an irreducible integer polynomial, then either   or  .
  • The Mahler measure of a monic integer polynomial is a Perron number.

Higher-dimensional Mahler measureEdit

The Mahler measure   of a multi-variable polynomial   is defined similarly by the formula[2]

 
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  -functions. For example, in 1981, Smyth[3] proved the formulas

 
where   is the Dirichlet L-function, and
 
where   is the Riemann zeta function. Here   is called the logarithmic Mahler measure.

Some results by Lawton and BoydEdit

From the definition, the Mahler measure is viewed as the integrated values of polynomials over the torus (also see Lehmer's conjecture). If   vanishes on the torus  , then the convergence of the integral defining   is not obvious, but it is known that   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   denote the integers and define   . If   is a polynomial in   variables and   define the polynomial   of one variable by

 

and define   by

 

where  .

Theorem (Lawton) — Let   be a polynomial in N variables with complex coefficients. Then the following limit is valid (even if the condition that   is relaxed):

 

Boyd's proposalEdit

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

 
where   is the m-th cyclotomic polynomial, the   are integers, and the   are chosen minimally so that   is a polynomial in the  . Let   be the set of polynomials that are products of monomials   and extended cyclotomic polynomials.

Theorem (Boyd) — Let   be a polynomial with integer coefficients. Then   if and only if   is an element of  .

This led Boyd to consider the set of values

 
and the union  . He made the far-reaching conjecture[5] that the set of   is a closed subset of  . 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   , Boyd further conjectures that
 

Mahler measure and entropyEdit

An action   of   by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module   over the ring  .[7] The topological entropy (which is equal to the measure-theoretic entropy) of this action,  , is given by a Mahler measure (or is infinite).[8] In the case of a cyclic module   for a non-zero polynomial   the formula proved by Lind, Schmidt, and Ward gives  , the logarithmic Mahler measure of  . 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   of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of   or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus   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 alsoEdit

NotesEdit

  1. ^ Although this is not a true norm for values of  .
  2. ^ Schinzel 2000, p. 224.
  3. ^ Smyth 2008.
  4. ^ Lawton 1983.
  5. ^ a b Boyd 1981a.
  6. ^ a b Boyd 1981b.
  7. ^ Kitchens, Bruce; Schmidt, Klaus (1989). "Automorphisms of compact groups". Ergodic Theory Dynam. 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". Invent. Math. 101: 593–629. doi:10.1007/BF01231517.
  9. ^ Lind, Douglas (1977). "The structure of skew products with ergodic group automorphisms". Israel J. Math. 28, no. 3: 205–248. doi:10.1007/BF02759810.

ReferencesEdit

  • 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, F. (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 (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
  • 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 linksEdit

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