Mahler’s Measure and the Dilogarithm (I)

By David W. Boyd, Fernando Rodriguez-Villegas. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2002

99 citations recorded by the source corpus.

Abstract An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,\,y)\,=\,P(x)y-q(x)$ , where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value ${{\text{ }\!\!\zeta\!\!\text{ }}_{F}}(2)$ for certain quadratic and quartic fields.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c6d39ca4af5ae455a8935af58cc457b97def0d21

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

On the Mahler Measure of 1+X+1/X+Y +1/Y (2011, background): “These volumes are well defined, and can be calculated in terms of values of the Bloch–Wigner dilogarithm [7].”
3-Manifolds and 3d Indices (2011, background): “om gluing two adjacent sides of a single tetrahedron together. Such triangulations are automatically \hard"; but more seriously, they fail to describe the moduli space of at connections on M(cf. [55], Sec. 10.3, and [56,57]), and should never be expected to produce the correct theory T M or its”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, background): “m matrices that are either upper or lower triangular | hence possibly parabolic | but never diagonal. This can be seen explicitly either from constructions involving three-dimensional developing maps [61, 64], or from more { 48 { combinatorial viewpoints as in [57]. In the case of a knot complement ”
Gauge Theories Labelled by Three-Manifolds (2011, background): “e-space is (P)SL(2;C), cf. [6, 7, 8, 9]. Almost all at connections can be realized as (possibly degenerate) hyperbolic metrics; for further remarks on this in the context of ideal triangulations, see [10, 11], and Section 4 of [12]. { 7 { This is a symplectic phase space, with a natural holomorphic ”
Number Theory and Polynomials: The Mahler measure of algebraic numbers: a survey (2007, unknown): “For this we refer the reader to [18], [33], [163], [39], [43], [40], [79, Chapter 3].”