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.

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 (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
[a3]	D.W. Boyd, "Mahler's measure and special values of -functions" Experim. Math. , 37 (1998) pp. 37–82
[a4]	C. Deninger, "Deligne periods of mixed motives, -theory and the entropy of certain -actions" J. Amer. Math. Soc. , 10 (1997) pp. 259–281
[a5]	E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial" Acta Arith. , 34 (1979) pp. 391–401
[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
[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
[a8]	G. Everest, Ni Fhlathúin Brid, "The elliptic Mahler measure" Math. Proc. Cambridge Philos. Soc. , 120 : 1 (1996) pp. 13–25
[a9]	K. Mahler, "On some inequalities for polynomials in several variables" J. London Math. Soc. , 37 : 2 (1962) pp. 341–344
[a10]	K. Schmidt, "Dynamical systems of algebraic origin" , Birkhäuser (1995)