Uniform Approximation to Mahler’s Measure in Several Variables

Boyd, David W.

Canadian mathematical bulletin, Tome 41 (1998) no. 1, pp. 125-128

Résumé

If $f({{x}_{1}},...,{{x}_{k}})$ 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 ${{\mathbb{T}}^{k}}$ . We construct a sequence of approximations ${{M}_{n}}\,(f)$ which satisfy $-d{{2}^{-n}}\,\log \,2\,+\,\log \,{{M}_{n}}(f)\,\le \,\log \,M(f)\,\le \,\log \,{{M}_{n}}(f)$ . 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.

Détail

DOI : 10.4153/CMB-1998-019-6

Mots-clés : 11R06, 11K16, 11Y99, Mahler measure, polynomials, computation