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Given a [[Polynomial|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|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*}

[[#References|[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 [[#References|[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|$K$-theory]]. For example, $m ( 1 + x + y ) = L ^ { \prime } ( - 1 , \chi _{- 3} )$, where $L ( s , \chi_{- 3} )$ is the [[Dirichlet L-function|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|elliptic curve]] of conductor $15$. This formula has not been proved but has been verified to over $50$ decimal places [[#References|[a3]]], [[#References|[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 [[#References|[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 [[#References|[a1]]]. An important open question, known as [[Lehmer conjecture|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|Pisot number]], the real root of $x ^ { 3 } - x - 1$ [[#References|[a6]]], [[#References|[a1]]]. A possible value for $c_0$ is $\operatorname { log } \sigma _ { 1 }$, where $\sigma _ { 1 } = 1.17628 \ldots$ is the smallest known [[Salem number|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 $P$ [[#References|[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 $m ( P ) \geq c _ { 2 } ( s )$ [[#References|[a7]]], [[#References|[a1]]].

A recent development is the elliptic Mahler measure [[#References|[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====
<table><tr><td valign="top">[a1]</td> <td valign="top"> D.W. Boyd, "Kronecker's theorem and Lehmer's problem for polynomials in several variables" ''J. Number Th.'' , '''13''' (1981) pp. 116–121</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> D.W. Boyd, "Two sharp inequalities for the norm of a factor of a polynomial" ''Mathematika'' , '''39''' (1992) pp. 341–349 {{MR|1203290}} {{ZBL|0758.30003}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> D.W. Boyd, "Mahler's measure and special values of $L$-functions" ''Experim. Math.'' , '''37''' (1998) pp. 37–82</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> 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 {{MR|1415320}} {{ZBL|}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial" ''Acta Arith.'' , '''34''' (1979) pp. 391–401 {{MR|0543210}} {{ZBL|0416.12001}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> 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 {{MR|0289451}} {{ZBL|0235.12003}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> E. Dobrowolski, "Mahler's measure of a polynomial in function of the number of its coefficients" ''Canad. Math. Bull.'' , '''34''' (1991) pp. 186–195 {{MR|1113295}} {{ZBL|}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> G. Everest, Ni Fhlathúin Brid, "The elliptic Mahler measure" ''Math. Proc. Cambridge Philos. Soc.'' , '''120''' : 1 (1996) pp. 13–25 {{MR|1373343}} {{ZBL|0865.11068}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> K. Mahler, "On some inequalities for polynomials in several variables" ''J. London Math. Soc.'' , '''37''' : 2 (1962) pp. 341–344 {{MR|0138593}} {{ZBL|0105.06301}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> K. Schmidt, "Dynamical systems of algebraic origin" , Birkhäuser (1995) {{MR|1345152}} {{ZBL|0833.28001}} </td></tr></table>