The Mahler measure of exact polynomials in three variables

Abstract

We prove that under certain explicit conditions, the Mahler measure of a three-variable polynomial can be expressed in terms of elliptic curve $L$-values and Bloch-Wigner dilogarithmmic values, conditionally on Beilinson's conjecture. In some cases, these dilogarithmic values simplify to Dirichlet $L$-values. The proof involves a construction of an element in $K_4^{(3)}$ of a smooth projective curve over a number field. This generalizes a result of Lalín for the polynomial $z+(x+1)(y+1)$. We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.

43 pages, accepted version, to appear in Algebra & Number Theory

References in corpus (1)

• An invariant property of Mahler measure