Given a polynomial with complex coefficients, the logarithmic Mahler measure is defined to be the average over the unit -torus of , i.e.
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
so that .
The Mahler measure is useful in the study of polynomial inequalities because of the multiplicative property . The important basic inequality
[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. 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.
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|[a3]||D.W. Boyd, "Mahler's measure and special values of -functions" Experim. Math. , 37 (1998) pp. 37–82|
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|[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|
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Mahler measure. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Mahler_measure&oldid=39967