An explicit formula for the Mahler measure of a family of 3-variable polynomials
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 683-700

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.

On montre une formule explicite pour la mesure de Mahler du polynôme a+bx -1 +cy+(a+bx+cy)z en termes de dilogarithmes et trilogarithmes.

@article{JTNB_2002__14_2_683_0,
     author = {Smyth, Chris J.},
     title = {An explicit formula for the Mahler measure of a family of $3$-variable polynomials},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {683-700},
     zbl = {1071.11018},
     mrnumber = {2040701},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_683_0}
}
Smyth, Chris J. An explicit formula for the Mahler measure of a family of $3$-variable polynomials. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 683-700. http://www.numdam.org/item/JTNB_2002__14_2_683_0/

[B1] D.W. Boyd, Speculations concerning the range of Mahler's measure. Canad. Math Bull. 24 (1981), 453-469. | MR 644535 | Zbl 0474.12005

[B2] D.W. Boyd, Mahler's measure and special values of L-functions. Experiment. Math. 7 (1998), 37-82. | MR 1618282 | Zbl 0932.11069

[B3] D.W. Boyd, Uniform approximation to Mahler's measure in several variables. Canad. Math. Bull. 41 (1998), 125-128. | MR 1618904 | Zbl 0898.11040

[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. | MR 1689496 | Zbl 0990.11061

[BRV] D.W. Boyd, F. Rodriguez Villegas, Mahler's measure and the dilogarithm. I. Canad. J. Math. 54 (2002), 468-492. | MR 1900760 | Zbl 1032.11028

[K] E.E. Kummer, Über die Transcendenten, welche aus wiederholten Integrntionen rationaler Formeln entstehen. J. für Math. (Crelle) 21 (1840), 328-371. | Zbl 021.0671cj

[L] L. Lewin, Dilogarithms and Associated Functions. Macdonald, London, 1958. | MR 105524 | Zbl 0083.35904

[R] G.A. Ray, Relations between Mahler's measure and values of L-series. Canad. J. Math. 39 (1987), 694-732. | MR 905752 | Zbl 0621.12005

[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. | MR 1691309 | Zbl 0980.11026

[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. | MR 1770638 | Zbl 0956.12001

[Sm] C.J. Smyth, On measures of polynomials in several variables. Bull. Austral. Math. Soc. Ser. A 23 (1981), 49-63. Corrigendum (with G. Myerson): Bull. Austral. Math. Soc. Ser. A 26 (1982), 317-319. | MR 615132 | Zbl 0442.10034

[Z] D. Zagier, The Bloch- Wigner-Ramakrishnan polylogarithm function. Math. Ann. 286 (1990), 613-624. | MR 1032949 | Zbl 0698.33001