The Mahler measure of exact polynomials in three variables

The Mahler measure of exact polynomials in three variables Trieu Thu HaWe 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 dilogarithmic 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 (3) 4 of a smooth projective curve over a number field.This generalizes a result of Laln (2015) for the polynomial z + (x + 1)(y + 1).We apply our method to several other Mahler measure identities conjectured by Boyd and Brunault.