Generalized Mahler measure of a family of polynomials
In this thesis we consider a variation of the Mahler measure where the defining integral is performed over a more general torus. Our work is based on a tempered family of polynomials originally studied by Boyd, Boyd P_k (x, y) = x + 1/x + y + 1/y + k with k ∈ R_{>4}. For the k = 4 case we use special values of the Bloch-Wigner dilogarithm to obtain the Mahler measure of P_4 over an arbitrary torus (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } with a, b ∈ R_{> 0}. Next we establish a relation between the Mahler measure of P_8 over a torus(T_ {a, √a} )^2 and its standard Mahler measure. The combination of this relation with results due to Lalin, Rogers, and Zudilin leads to a formula involving the generalized Mahler measure of this polynomial given in terms of L'(E, 0). In the end, we propose a strategy to prove some similar results for the general case k > 4 over (T_ {a, b})^2 with some restrictions on a, b.
Mahler measure Bloch-Wigner dilogarithm L-functions of elliptic curves Arbitrary torus Regulator