Uniform Approximation to Mahler’s Measure in Several Variables

Abstract

AbstractIf ƒ (x1,…,xk) is a polynomial with complex coefficients, the Mahler measure of ƒ , M(ƒ) is defined to be the geometric mean of |f| over the k-torus 𝕋k. We construct a sequence of approximations Mn(ƒ) which satisfy −d2−n log 2 + log Mn(ƒ) ≤ log M(ƒ) ≤ log Mn(ƒ). We use these to prove that M(ƒ) is a continuous function of the coefficients of ƒ for polynomials of fixed total degree d. Since Mn(ƒ) can be computed in a finite number of arithmetic operations from the coefficients of ƒ this also demonstrates an effective (but impractical) method for computing M(ƒ) to arbitrary accuracy.