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# Uniform approximation to Mahler's measure in several variables

If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus $\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy $-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.