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Uniform approximation to Mahler's measure in several variables

  Published:1998-03-01
 Printed: Mar 1998
  • David W. Boyd
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Abstract

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.
MSC Classifications: 11R06, 11K16, 11Y99 show english descriptions PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
None of the above, but in this section
11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11K16 - Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
11Y99 - None of the above, but in this section
 

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