An explicit formula for the Mahler measure of a family of 3 -variable polynomials

Chris J. Smyth

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 683-700
  • ISSN: 1246-7405

Abstract

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An explicit formula for the Mahler measure of the 3 -variable Laurent polynomial a + b x - 1 + c y + ( a + b x + c y ) z is given, in terms of dilogarithms and trilogarithms.

How to cite

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Smyth, Chris J.. "An explicit formula for the Mahler measure of a family of $3$-variable polynomials." Journal de théorie des nombres de Bordeaux 14.2 (2002): 683-700. <http://eudml.org/doc/248892>.

@article{Smyth2002,
abstract = {An explicit formula for the Mahler measure of the $3$-variable Laurent polynomial $a + bx^\{-1\} + cy + (a + bx + cy)z$ is given, in terms of dilogarithms and trilogarithms.},
author = {Smyth, Chris J.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Mahler's measure; polylogarithm},
language = {eng},
number = {2},
pages = {683-700},
publisher = {Université Bordeaux I},
title = {An explicit formula for the Mahler measure of a family of $3$-variable polynomials},
url = {http://eudml.org/doc/248892},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Smyth, Chris J.
TI - An explicit formula for the Mahler measure of a family of $3$-variable polynomials
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 683
EP - 700
AB - An explicit formula for the Mahler measure of the $3$-variable Laurent polynomial $a + bx^{-1} + cy + (a + bx + cy)z$ is given, in terms of dilogarithms and trilogarithms.
LA - eng
KW - Mahler's measure; polylogarithm
UR - http://eudml.org/doc/248892
ER -

References

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  1. [B1] D.W. Boyd, Speculations concerning the range of Mahler's measure. Canad. Math Bull.24 (1981), 453-469. Zbl0474.12005MR644535
  2. [B2] D.W. Boyd, Mahler's measure and special values of L-functions. Experiment. Math.7 (1998), 37-82. Zbl0932.11069MR1618282
  3. [B3] D.W. Boyd, Uniform approximation to Mahler's measure in several variables. Canad. Math. Bull.41 (1998), 125-128. Zbl0898.11040MR1618904
  4. [B4] D.W. Boyd, Mahler's measure and special values of L-functions-some conjectures. Number theory in progress, Vol. 1 (Zakopane-Koscielisko, 1997), 27-34, de Gruyter, Berlin, 1999. Zbl0990.11061MR1689496
  5. [BRV] D.W. Boyd, F. Rodriguez Villegas, Mahler's measure and the dilogarithm. I. Canad. J. Math.54 (2002), 468-492. Zbl1032.11028MR1900760
  6. [K] E.E. Kummer, Über die Transcendenten, welche aus wiederholten Integrntionen rationaler Formeln entstehen. J. für Math. (Crelle) 21 (1840), 328-371. 
  7. [L] L. Lewin, Dilogarithms and Associated Functions. Macdonald, London, 1958. Zbl0083.35904MR105524
  8. [R] G.A. Ray, Relations between Mahler's measure and values of L-series. Canad. J. Math.39 (1987), 694-732. Zbl0621.12005MR905752
  9. [RV] F. Rodriguez Villegas, Modular Mahler measures. I, Topics in number theory (University Park, PA, 1997), 17-48, Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999. Zbl0980.11026MR1691309
  10. [Sc] A. Schinzel, Polynomials with Special Regard to Reducibility. With an appendix by Umberto Zannier. Encyclopedia of Mathematics and its Applications, 77, Cambridge University Press, Cambridge, 2000. Zbl0956.12001MR1770638
  11. [Sm] C.J. Smyth, On measures of polynomials in several variables. Bull. Austral. Math. Soc. Ser. A23 (1981), 49-63. Corrigendum (with G. Myerson): Bull. Austral. Math. Soc. Ser. A26 (1982), 317-319. Zbl0442.10034MR615132
  12. [Z] D. Zagier, The Bloch- Wigner-Ramakrishnan polylogarithm function. Math. Ann.286 (1990), 613-624. Zbl0698.33001MR1032949

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