Mahler measure

Given a polynomial $P ( x _ { 1 } , \ldots , x _ { n } )$ with complex coefficients, the logarithmic Mahler measure $m ( P )$ is defined to be the average over the unit $n$-torus of $\operatorname { log } | P ( x _ { 1 } , \dots , x _ { n } ) |$, i.e.

\begin{equation*} m ( P ) = \int _ { 0 } ^ { 1 } \ldots \int _ { 0 } ^ { 1 } \operatorname { log } | P ( e ^ { i t_{1} } , \ldots , e ^ { i t _ { n } } ) | d t _ { 1 } \ldots d t _ { n }. \end{equation*}

The Mahler measure is defined by $M ( P ) = \operatorname { exp } ( m ( P ) )$, so that $M ( P )$ is the geometric mean of $| P |$ over the $n$-torus. If $n = 1$ and $P ( x ) = a _ { 0 } \prod _ { k = 1 } ^ { d } ( x - \alpha _ { k } )$, Jensen's formula gives the explicit formula

\begin{equation*} m ( P ) = \operatorname { log } | a _ { 0 } | + \sum _ { k = 1 } ^ { d } \operatorname { log } ( \operatorname { max } ( | \alpha _ { k } | , 1 ) ), \end{equation*}

so that $M ( P ) = | a _ { 0 } | \prod _ { k = 1 } ^ { d } \operatorname { max } ( | \alpha _ { k } | , 1 )$.

The Mahler measure is useful in the study of polynomial inequalities because of the multiplicative property $M ( P Q ) = M ( P ) M ( Q )$. The important basic inequality

\begin{equation*} M ( P ) \leq L ( P ) \leq 2 ^ { d } M ( P ) \end{equation*}

[a9] relates $M ( P )$ to $L ( P )$, the sum of the absolute values of the coefficients of $P$, where $d$ denotes the total degree of $P$, i.e. the sum of the degrees in each variable separately. A recent inequality for polynomials of one variable is that $\| P \| _{\infty} \| Q \| _{\infty} \leq \delta^{d} \| PQ \| _{\infty} $, where $\| P \| _ { \infty } = \operatorname { max } _ { | z | = 1 } | P ( z ) |$, $d$ is the sum of the degrees of $P$ and $Q$, and $\delta = M ( 1 + x + y - x y ) = 1.7916228\dots$ is the best possible constant [a2].

Specializing to polynomials with integer coefficients, in case $n = 1$, $m ( P )$ is the logarithm of an algebraic integer. If $n > 1$, there are few explicit formulas known, but those that do exist suggest that $m ( P )$ has intimate connections with $K$-theory. For example, $m ( 1 + x + y ) = L ^ { \prime } ( - 1 , \chi _{- 3} )$, where $L ( s , \chi_{- 3} )$ is the Dirichlet $L$-function for the odd primitive character of conductor $3$, i.e. $\chi_{ - 3} ( n ) = \left( \frac { - 3 } { n } \right)$, and it has been conjectured that $m ( x + y + x y + x ^ { 2 } y + x y ^ { 2 } ) = L ^ { \prime } ( 0 , E _ { 15 } )$, where $L ( s , E _ { 15 } )$ is the $L$-function of an elliptic curve of conductor $15$. This formula has not been proved but has been verified to over $50$ decimal places [a3], [a4].

The Mahler measure $m ( P )$ occurs naturally as the growth rate in many problems, for example as the entropy of certain ${\bf Z} ^ { d }$-actions [a10]. The set of $P$ for which $m ( P ) = 0$ is known: in case $n = 1$, a theorem of Kronecker shows that these are products of cyclotomic polynomials and monomials. In case $n > 1$, these are the generalized cyclotomic polynomials [a1]. An important open question, known as Lehmer's problem, is whether there is a constant $c_0 > 0$ such that if $m ( P ) > 0$, then $m ( P ) \geq c_0$. This is known to be the case if $P$ is a non-reciprocal polynomial, where a polynomial is reciprocal if $P ( x _ { 1 } ^ { - 1 } , \ldots , x _ { n } ^ { - 1 } ) / P ( x _ { 1 } , \ldots , x _ { n } )$ is a monomial. In this case, $m ( P ) \geq \operatorname { log } \theta _ { 0 }$, where $\theta _ { 0 } = 1.3247 \ldots > 1$ is the smallest Pisot number, the real root of $x ^ { 3 } - x - 1$ [a6], [a1]. A possible value for $c_0$ is $\operatorname { log } \sigma _ { 1 }$, where $\sigma _ { 1 } = 1.17628 \ldots$ is the smallest known Salem number, a number of degree $10$ known as Lehmer's number.

For $n = 1$, the best result in this direction is that $m ( P ) > c _ { 1 } ( \operatorname { log } \operatorname { log } d / \operatorname { log } d ) ^ { 3 }$, where $c_1$ is an explicit absolute constant and $d$ is the degree of $P$ [a5]. A result that applies to polynomials in any number of variables is an explicit constant $c _ { 2 } ( s ) > 0$ depending on the number $s$ of non-zero coefficients of $P$ such that $m ( P ) \geq c _ { 2 } ( s )$ [a7], [a1].

A recent development is the elliptic Mahler measure [a8], in which the torus $\bf T$ 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 $L$-functions" Experim. Math. , 37 (1998) pp. 37–82
[a4]	C. Deninger, "Deligne periods of mixed motives, $K$-theory and the entropy of certain $\mathbf{Z} ^ { n }$-actions" J. Amer. Math. Soc. , 10 (1997) pp. 259–281 MR1415320
[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