The Mahler measure M(p) of a multi-variable polynomial p(x_1,\ldots,x_n) \in \mathbb{C}[x_1,\ldots,x_n] is defined similarly by the formula M(p) = \exp\left(\int_0^{1} \int_0^{1} \cdots \int_0^{1} \log \Bigl( \bigl |p(e^{2\pi i\theta_1}, e^{2\pi i\theta_2}, \ldots, e^{2\pi i\theta_n}) \bigr| \Bigr) \, d\theta_1\, d\theta_2\cdots d\theta_n \right). It inherits the above three properties of the Mahler measure for a one-variable polynomial. The multi-variable Mahler measure has been shown, in some cases, to be related to special values of
zeta-functions and
L-functions. For example, in 1981, Smyth proved the formulas m(1+x+y)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2) where L(\chi_{-3},s) is a
Dirichlet L-function, and m(1+x+y+z)=\frac{7}{2\pi^2}\zeta(3), where \zeta is the
Riemann zeta function. Here m(P)=\log M(P) is called the
logarithmic Mahler measure.
Some results by Lawton and Boyd From the definition, the Mahler measure is viewed as the integrated values of polynomials over the torus (also see
Lehmer's conjecture). If p vanishes on the torus (S^1)^n, then the convergence of the integral defining M(p) is not obvious, but it is known that M(p) does converge and is equal to a limit of one-variable Mahler measures, which had been conjectured by
Boyd. This is formulated as follows: Let \mathbb{Z} denote the integers and define \mathbb{Z}^N_+=\{r=(r_1,\dots,r_N)\in\mathbb{Z}^N:r_j\ge0\ \text{for}\ 1\le j\le N\} . If Q(z_1,\dots,z_N) is a polynomial in N variables and r=(r_1,\dots,r_N)\in\mathbb{Z}^N_+ define the polynomial Q_r(z) of one variable by Q_r(z):=Q(z^{r_1},\dots,z^{r_N}) and define q(r) by q(r) := \min \left\{H(s):s=(s_1,\dots,s_N)\in\mathbb{Z}^N, s\ne(0,\dots,0)~\text{and}~\sum^N_{j=1} s_j r_j = 0 \right\} where H(s)=\max\{|s_j|:1\le j\le N\}. {{math theorem | name = Theorem (Lawton) | math_statement = Let Q(z_1,\dots,z_N) be a polynomial in
N variables with complex coefficients. Then the following limit is valid (even if the condition that r_i \geq 0 is relaxed): \lim_{q(r)\to\infty}M(Q_r)=M(Q)}}
Boyd's proposal Boyd provided more general statements than the above theorem. He pointed out that the classical
Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk, can be regarded as characterizing those polynomials of one variable whose measure is exactly 1, and that this result extends to polynomials in several variables. The
topological entropy (which is equal to the
measure-theoretic entropy) of this action, h(\alpha_N), is given by a Mahler measure (or is infinite). In the case of a
cyclic module M=R/\langle F\rangle for a non-zero polynomial F(z_1,\dots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n] the formula proved by Lind,
Schmidt, and
Ward gives h(\alpha_N)=\log M(F), the logarithmic Mahler measure of F. In the general case, the entropy of the action is expressed as a sum of logarithmic Mahler measures over the generators of the
principal associated prime ideals of the module. As pointed out earlier by Lind in the case n=1 of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of [0,\infty] or a countable set depending on the solution to
Lehmer's problem. Lind also showed that the infinite-dimensional torus \mathbb{T}^{\infty} either has
ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem. == See also ==