Measure of polynomial height
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.
Properties
- 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 measure
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 formulaswhere is a Dirichlet L-function, andwhere is the Riemann zeta function. Here 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 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 .
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 formwhere 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.
This led Boyd to consider the set of valuesand 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 entropy
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 also
Notes
- ^ Although this is not a true norm for values of .
- ^ Schinzel 2000, p. 224.
- ^ Smyth 2008.
- ^ Lawton 1983.
- ^ a b Boyd 1981a.
- ^ a b 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.
References
- 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
- 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.
- 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