Mahler measure
Given a polynomial with complex coefficients, the logarithmic Mahler measure
is defined to be the average over the unit
-torus of
, i.e.
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The Mahler measure is defined by , so that
is the geometric mean of
over the
-torus. If
and
, Jensen's formula gives the explicit formula
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so that .
The Mahler measure is useful in the study of polynomial inequalities because of the multiplicative property . The important basic inequality
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[a9] relates to
, the sum of the absolute values of the coefficients of
, where
denotes the total degree of
, i.e. the sum of the degrees in each variable separately. A recent inequality for polynomials of one variable is that
, where
,
is the sum of the degrees of
and
, and
is the best possible constant [a2].
Specializing to polynomials with integer coefficients, in case ,
is the logarithm of an algebraic integer (cf. Algebraic number). If
, there are few explicit formulas known, but those that do exist suggest that
has intimate connections with
-theory. For example,
, where
is the Dirichlet
-function for the odd primitive character of conductor
, i.e.
, and it has been conjectured that
, where
is the
-function of an elliptic curve of conductor
. This formula has not been proved but has been verified to over
decimal places [a3], [a4].
The Mahler measure occurs naturally as the growth rate in many problems, for example as the entropy of certain
-actions [a10]. The set of
for which
is known: in case
, a theorem of Kronecker shows that these are products of cyclotomic polynomials and monomials. In case
, these are the generalized cyclotomic polynomials [a1]. An important open question, known as Lehmer's problem, is whether there is a constant
such that if
, then
. This is known to be the case if
is a non-reciprocal polynomial, where a polynomial is reciprocal if
is a monomial. In this case,
, where
is the smallest Pisot number, the real root of
[a6], [a1]. A possible value for
is
, where
is the smallest known Salem number, a number of degree
known as Lehmer's number.
For , the best result in this direction is that
, where
is an explicit absolute constant and
is the degree of
[a5]. A result that applies to polynomials in any number of variables is an explicit constant
depending on the number
of non-zero coefficients of
such that
[a7], [a1].
A recent development is the elliptic Mahler measure [a8], in which the torus is replaced by an elliptic curve. It seems likely that this will have an interpretation as the entropy of a dynamical system but this remains as of yet (1998) a future development.
References
[a1] | D.W. Boyd, "Kronecker's theorem and Lehmer's problem for polynomials in several variables" J. Number Th. , 13 (1981) pp. 116–121 |
[a2] | D.W. Boyd, "Two sharp inequalities for the norm of a factor of a polynomial" Mathematika , 39 (1992) pp. 341–349 MR1203290 Zbl 0758.30003 |
[a3] | D.W. Boyd, "Mahler's measure and special values of ![]() |
[a4] | C. Deninger, "Deligne periods of mixed motives, ![]() ![]() |
[a5] | E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial" Acta Arith. , 34 (1979) pp. 391–401 MR0543210 Zbl 0416.12001 |
[a6] | C.J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer" Bull. London Math. Soc. , 3 (1971) pp. 169–175 MR0289451 Zbl 0235.12003 |
[a7] | E. Dobrowolski, "Mahler's measure of a polynomial in function of the number of its coefficients" Canad. Math. Bull. , 34 (1991) pp. 186–195 MR1113295 |
[a8] | G. Everest, Ni Fhlathúin Brid, "The elliptic Mahler measure" Math. Proc. Cambridge Philos. Soc. , 120 : 1 (1996) pp. 13–25 MR1373343 Zbl 0865.11068 |
[a9] | K. Mahler, "On some inequalities for polynomials in several variables" J. London Math. Soc. , 37 : 2 (1962) pp. 341–344 MR0138593 Zbl 0105.06301 |
[a10] | K. Schmidt, "Dynamical systems of algebraic origin" , Birkhäuser (1995) MR1345152 Zbl 0833.28001 |
Mahler measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mahler_measure&oldid=18597