Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T20:57:06.138Z Has data issue: false hasContentIssue false
  • English
  • Français

Mahler’s Measure and the Dilogarithm (I)

Published online by Cambridge University Press:  20 November 2018

David W. Boyd  and
Fernando Rodriguez-Villegas
David W. Boyd
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, e-mail: boyd@math.ubc.ca
Fernando Rodriguez-Villegas
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas, 78712 USA, e-mail: villegas@math.utexas.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

11G4011R0611Y35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 3 , 01 June 2002 , pp. 468 - 492
Copyright
Copyright © Canadian Mathematical Society 2002

References

[Bo1] Boyd, D. W., Speculations concerning the range of Mahler's measure. Canad. Math. Bull. 24 (1981), 453–469.Google Scholar
[Bo2] Boyd, D. W., Two sharp inequalities for the norm of a factor of a polynomial. Mathematika 39 (1992), 341–349.Google Scholar
[Bo3] Boyd, D. W., Mahler's Measure and Special Values of L-functions. Experiment. Math. 37 (1998), 37–82.Google Scholar
[Bo4] Boyd, D. W., Mahler's measure and special Values of L-functions—some conjectures. In: Number Theory in Progress 1, (eds., K. Györy, H. Iwaniec and J. Urbanowicz), de Gruyter, Berlin, 1999, 27–34.Google Scholar
[Bo5] Boyd, D. W., Mahler's measure and invariants of hyperbolic manifolds. In: Number Theory for the Millennium, (ed., B. C. Berndt et al.), A. K. Peters, Boston, 2002.Google Scholar
[Br] Browkin, J., Conjectures on the Dilogarithm. K-Theory 3 (1989), 29–56.Google Scholar
[Ch] Chinburg, T., Mahler measures and derivatives of L-functions at non-positive integers. 1984, preprint.Google Scholar
[HW] Hildebrand, M. and Weeks, J., A computer generated census of cusped hyperbolic 3-manifolds. In: Computers and Mathematics, (eds., E. Kaltofen and S.Watts), Springer-Verlag, New York, 1989, 53–59.Google Scholar
[La] Lalande, F., Corps de nombres engendrés par un nombre de Salem. Acta Arith. 88 (1999), 191–200.Google Scholar
[Le] Lewin, L., Polylogarithms and associated functions. North Holland, 1981.Google Scholar
[MPV] Mossinghoff, M. J., Pinner, C. G. and Vaaler, J. D., Perturbing polynomials with all their roots on the unit circle. Math. Comp. 67 (1998), 1707–1726.Google Scholar
[Ra] Ray, G. A., Relations between Mahler's measure and values of L-series. Canad. J. Math. 39 (1987), 694–732.Google Scholar
[RV] Rodriguez Villegas, F., Modular Mahler measures I. Topics in Number Theory, (eds., S. D. Ahlgren, G. E. Andrews and K. Ono), Kluwer, Dordrecht, 1999, 17–48.Google Scholar
[Sc] Schinzel, A., Primitive divisors of the expression An − Bn in algebraic number fields. J. Reine Angew. Math. (9) 268 (1974), 27–33.Google Scholar
[Sm] Smyth, C. J., On measures of polynomials in several variables. Bull. Austral. Math. Soc. 23 (1981), 49–63.Google Scholar
[Za1] Zagier, D., The Bloch-Wigner-Ramakrishnan polylogarithm function. Math. Ann. 286 (1990), 613–624.Google Scholar
[Za2] Zagier, D., Special Values and Functional Equations of Polylogarithms. In: Structural Properties of Polylogarithms, (ed., L. Lewin), Amer. Math. Soc., Providence, 1991, 377–400.Google Scholar