References

Some references on Mahler's measure, Lehmer's conjecture, and small Salem numbers.

Suggested additions are welcome.

Books

  1. M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J. P. Schreiber, Pisot and Salem Numbers, Berkhäuser, Basel, 1992.
  2. P. Borwein, Computational Excursions in Analysis and Number Theory, CMS Books Math. 10, Springer Verlag, 2002.
  3. G. Everest and T. Ward, Heights of Polynominals and Entropy in Algebraic Dynamics, Springer Verlag, 1999.
  4. K. Mahler, Lectures on Transcendental Numbers, Lecture Notes in Math. 546, Springer Verlag, 1976.
  5. M. Mignotte, Mathematics for Computer Algebra, Springer Verlag, 1992.
  6. M. Mignotte and D. Stefanescu, Polynomials: An Algorithmic Approach, Springer Verlag, 1999.
  7. A. Schinzel, Selected Topics on Polynomials, Univ. Michigan Press, 1982.
  8. A. Schinzel, Polynomials with Special Regard to Reducibility, Encyclopedia Math. Appl. 77, Cambridge Univ. Press, 2000.
  9. K. Schmidt, Dynamical Systems of Algebraic Origin, Progr. Math. 128, Birkhäuser, Basel, 1995.
  10. M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Grundlehren Math. Wiss. 326, Springer Verlag, Berlin, 2000.

Articles

  1. Mahler's Measure of Polynomials in One Variable.

    1. Lower Bounds: General Case.

      1. P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355-369.
      2. D. C. Cantor and E. G. Strauss, On a question of D. H. Lehmer, Acta Arith. 42 (1982), 97-100. Correction, ibid. 42 (1983), 327.
      3. E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401.
      4. R. Louboutin, Sur la mesure de Mahler d'un nombre algébrique, C. R. Acad. Sci. Paris 296 (1983), 707-708.
      5. E. M. Matveev, A relationship between the Mahler measure and the discriminant of algebraic numbers (Russian), Mat. Zametki 59 (1996) 415-420; translation in Math. Notes 59 (1996), 293-297.
      6. M. Mignotte, Entiers algébriques dont les conjugués sont proches du cercle unité, Séminaire Delange-Pisot-Poitou, 19e année: 1977/78, Théorie des nombres, Fasc. 2, Exp. No. 39, 6 pp.
      7. U. Rausch, On a theorem of Dobrowolski about the product of conjugate numbers, Colloq. Math. 50 (1985), 137-142.
      8. C. L. Stewart, Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. Math. France 106 (1978), 169-176.
      9. P. Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), 81-95.

    2. Lower Bounds: Special Cases Based on Locations of Roots.

      1. R. Breusch, On the distribution of the roots of a polynomial with integral coefficients, Proc. Amer. Math. Soc. 2 (1951), no. 6, 939-941.
      2. A. Dubickas, On the measure of a nonreciprocal algebraic number, Ramanujan J. 4 (2000), no. 3, 291-298.
      3. A. Dubickas and C. J. Smyth, The Lehmer constants of an annulus, J. Théor. Nombres Bordeaux 13 (2001), no. 2, 413-420.
      4. V. Flammang, Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers, Math. Comp. 65 (1996), no. 213, 307-311.
      5. V. Flammang, Inégalités sur la mesure de Mahler d'un pôlynome, J. Théor. Nombres Bordeaux 9 (1997), 69-74.
      6. J. Garza, On the height of algebraic numbers with real conjugates, Acta Arith. 124 (2007), no. 4, 385-389.
      7. G. Höhn and N.-P. Skoruppa, Un résultat de Schinzel, J. Théor. Nombres Bordeaux 5 (1993), no. 1, 185.
      8. M. Langevin, Méthode de Fekete-Szegö et problème de Lehmer, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 10, 463-466.
      9. M. Langevin, Minorations de la maison et de la mesure de Mahler de certains entiers algébriques, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 12, 523-526.
      10. M. Langevin, Calculs explicites de constantes de Lehmer, in Groupe de travail en théorie analytique et élémentaire des nombres, 1986-1987, Publ. Math. Orsay 88 (1988), Univ. Parix XI, Orsay, pp. 52-68.
      11. M. Mignotte, Sur un théorème de M. Langevin, Acta Arith. 54 (1989), no. 1, 81-86.
      12. L. Panaitopol, Minorations pour les mesures de Mahler de certains polynômes particuliers, J. Théor. Nombres Bordeaux 12 (2000), no. 1, 127-132.
      13. G. Rhin and C. J. Smyth, On the absolute Mahler measure of polynomials having all zeros in a sector, Math. Comp. 64 (1995), no. 209, 295-304.
      14. G. Rhin and C. J. Smyth, On the Mahler measure of the composition of two polynomials, Acta Arith. 79 (1997), no. 3, 239-247.
      15. G. Rhin and Q. Wu, On the absolute Mahler measure of polynomials having all zeros in a sector II, Math. Comp. 74 (2005), no. 249, 383-388.
      16. A. Schinzel, On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith. 24 (1973), 385-389. Addendum, 26 (1974/75), no. 3, 329-331.
      17. C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. Lond. Math. Soc. 3 (1971), 169-175.
      18. C. J. Smyth, On the measure of totally real algebraic integers, J. Austral. Math. Soc. Ser. A 30 (1980), no. 2, 137-149.
      19. C. J. Smyth, On the measure of totally real algebraic integers II, Math. Comp. 37 (1981), no. 155, 205-208.

    3. Lower Bounds: Special Cases Based on Coefficients.

      1. N. C. Bonciocat, Congruences and Lehmer's problem, Int. J. Number Theory 4 (2008), no. 4, 587-596.
      2. P. Borwein, K. G. Hare, and M. J. Mossinghoff, The Mahler measure of polynomials with odd coefficients, Bull. London Math. Soc. 36 (2004), no. 3, 332-338.
      3. P. Borwein, E. Dobrowolski, and M. J. Mossinghoff, Lehmer's problem for polynomials with odd coefficients, Ann. of Math. (2) 166 (2007), no. 2, 347-366.
      4. E. Dobrowolski, On a question of Lehmer, Mém. Soc. Math. France (N.S.) (1980/81), no. 2, 35-39.
      5. E. Dobrowolski, W. Lawton, and A. Schinzel, On a problem of Lehmer, in Studies in Pure Mathematics, Birkhäuser, Basel, 1983, pp. 135-144.
      6. E. Dobrowolski, Mahler's measure of a polynomial in function of the number of its coefficients, Canad. Math. Bull. 34 (1991), no. 2, 186-195.
      7. E. Dobrowolski, Mahler's measure of a polynomial in terms of the number of its monomials, Acta Arith. 123 (2006), no. 3, 201-231.
      8. A. Dubickas and M. J. Mossinghoff, Auxiliary polynomials for some problems regarding Mahler's measure, Acta Arith. 119 (2005), no. 1, 65-79.
      9. J. Garza, M. I. M. Ishak, M. J. Mossinghoff, C. Pinner, and B. Wiles, Heights of roots of polynomials with odd coefficients, preprint.
      10. C. L. Samuels, The Weil height in terms of an auxiliary polynomial, Acta Arith. 128 (2007), no. 3, 209-221.

    4. Lower Bounds: Special Cases Based on Algebraic Properties.

      1. F. Amoroso and R. Dvornicich, A lower bound for the height in abelian extensions, J. Number Theory 80 (2000), 260-272.
      2. F. Amoroso and S. David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179.
      3. E. Dobrowolski, A note on integer symmetric matrices and Mahler's measure, Canad. Math. Bull. 51 (2008), no. 1, 57-59.
      4. J. Garza, The Mahler measure of dihedral extensions, Acta Arith. 131 (2008), no. 3, 201-215.
      5. M. I. M. Ishak, M. J. Mossinghoff, C. Pinner, and B. Wiles, Lower bounds for heights in cyclotomic extensions, preprint.

    5. Computations.

      1. D. W. Boyd, Reciprocal polynomials having small Mahler measure, Math. Comp. 35 (1980), 1361-1377.
      2. D. W. Boyd, Reciprocal polynomials having small Mahler measure II, Math. Comp. 53 (1989), 355-357, S1-S5.
      3. L. Cerlienco, M. Mignotte, and F. Piras, Computing the measure of a polynomial, J. Symbolic Comput. 4 (1987), 21-33.
      4. V. Flammang, G. Rhin, and J.-M. Sac-Épée, Integer transfinite diameter and polynomials with small Mahler measure, Math. Comp. 75 (2006), no. 255, 1527-1540.
      5. M. J. Mossinghoff, Polynomials with small Mahler measure, Math. Comp. 67 (1998), 1697-1705, S11-S14.
      6. M. J. Mossinghoff, C. G. Pinner, and J. D. Vaaler, Perturbing polynomials with all their roots on the unit circle, Math. Comp. 67 (1998), 1707-1726.
      7. M. J. Mossinghoff, G. Rhin, and Q. Wu, Minimal Mahler measures, Experiment. Math., 17 (2008), no. 4, 451-458.
      8. G. A. Ray, A locally parameterized version of Lehmer's problem, in Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics, ed. by W. Gautschi, Proc. Symp. in Appl. Math. 48, Amer. Math. Soc., 1994, pp. 573-576.
      9. G. Rhin and J.-M. Sac-Épée, New methods providing high degree polynomials with small Mahler measure, Experiment. Math. 12 (2003), no. 4, 457-461.

    6. Values of Mahler's Measure (expected value, distribution, inverse problems).

      1. D. W. Boyd, Inverse problems for Mahler's measure, pp. 147-158 in Diophantine Analysis, ed. by J. H. Loxton and A. J. van der Poorten, London Math. Soc. Lecture Note Ser. 109, 1986.
      2. D. W. Boyd, Perron units which are not Mahler measures, Ergodic Theory Dynam. Systems 6 (1986), 485-488.
      3. D. W. Boyd, Reciprocal algebraic integers whose Mahler measures are nonreciprocal, Canad. Math. Bull. 30 (1987), 3-8.
      4. S.-J. Chern and J. D. Vaaler, The distribution of values of Mahler's measure, J. Reine Angew. Math. 540 (2001), 1-47.
      5. J. D. Dixon and A. Dubickas, The values of Mahler measures, Mathematika 51 (2004), 131-148.
      6. P. Drungilas and A. Dubickas, Every real algebraic integer is a difference of two Mahler measures, Canad. Math. Bull. 50 (2007), no. 2, 191-195.
        7. A. Dubickas and C. J. Smyth, On the metric Mahler measure, J. Number Theory 86 (2001), no. 2, 368-387.
      7. A. Dubickas, Some Diophantine properties of the Mahler measure (Russian), Mat. Zametki 72 (2002), no. 6, 828-833. English translation in Math. Notes 72 (2002), no. 5-6, 763-767.
      8. A. Dubickas, Mahler measures generate the largest possible groups, Math. Res. Lett. 11 (2004), no. 2-3, 279-283.
      9. A. Dubickas, Nonreciprocal algebraic numbers of small measure, Comment. Math. Univ. Carolin. 45 (2004), no. 4, 693-697.
      10. A. Dubickas, On numbers which are Mahler measures, Monatsh. Math. 141 (2004), no. 2, 119-126.
      11. A. Dubickas, Mahler measures in a cubic field, Czechoslovak Math. J. 56(131) (2006), no. 3, 949-956.
      12. A. Dubickas, Mahler measures in a field are dense modulo 1, Arch. Math. (Basel) 88 (2007), no. 1, 29-34.
      13. G. T. Fielding, The expected value of the integral around the unit circle of a certain class of polynomials, Bull. London Math. Soc. 2 (1970), 301-306.
      14. A. Schinzel, On values of the Mahler measure in a quadratic field, Acta Arith. 113 (2004), no. 4, 401-408.
      15. C. Sinclair, The distribution of Mahler's measures of reciprocal polynomials, Int. J. Math. Math. Sci. (2004), no. 49-52, 2773-2786.
      16. C. Sinclair, The range of multiplicative functions on C[x], R[x], and Z[x], Proc. Lond. Math. Soc. (3) 96 (2008), no. 3, 697-737.

    7. Other Inequalities (connections with Lp norms, etc.).

      1. F. Amoroso, Sur des polynômes des petites mesures de Mahler, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 1, 11-14.
      2. E. Beller and D. J. Newman, An extremal problem for the geometric mean of polynomials, Proc. Amer. Math. Soc. 39 (1973), 313-317.
      3. P. Borwein, M. J. Mossinghoff, and J. D. Vaaler, Generalizations of Gonçalves' inequality, Proc. Amer. Math. Soc. 135 (2007), no. 1, 253-261.
      4. J. Dégot, Finite-dimensional Mahler measure of a polynomial and Szegö's theorem, J. Number Theory 62 (1997), no. 2, 422-427.
      5. J. Dégot and O. Jenvrin, Bombieri's norm versus Mahler's measure, Illinois J. Math. 42 (1998), no. 2, 187-197.
      6. A. Durand, On Mahler's measure of a polynomial, Proc. Amer. Math. Soc. 83 (1981), no. 1, 75-76.
      7. T. Erdélyi and D. S. Lubinsky, Large sieve inequalities via subharmonic methods and the Mahler measure of the Fekete polynomials, Canad. J. Math. 59 (2007), no. 4, 730-741.
      8. V. Flammang, Comparaison de deux mesures de polynômes, Canad. Math. Bull. 38 (1995), no. 4, 438-444.
      9. J. V. Gonçalves, L'inégalité de W. Specht, Univ. Lisboa. Revista Fac. Ci. A. Ci. Mat. (2) 1, (1950), 167-171.
      10. W. Lawton, Heights of algebraic numbers and Szegö's theorem, Proc. Amer. Math. Soc. 49 (1975), 47-50.
      11. K. Mahler, An application of Jensen's formula to polynomials, Mathematika 7 (1960), 98-100.
      12. K. Mahler, On the zeros of the derivative of a polynomial, Proc. Roy. Soc. Ser. A 264 (1961), 145-154.
      13. K. Mahler, On two extremum properties of polynomials, Illinois J. Math. 7 (1963), 681-701.
      14. K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257-262.
      15. K. Mahler, A remark on a paper of mine on polynomials, Illinois J. Math. 8 (1964), 1-4.
      16. M. Mignotte, An inequality about irreducible factors of integer polynomials, J. Number Theory 30 (1988), no. 2, 156-166.
      17. M. Mignotte, An inequality about factors of polynomials, Math. Comp. 28 (1974), 1153-1157.
      18. A. M. Ostrowski, On an inequality of J. Vicente Gonçalves, Univ. Lisboa Revista Fac. Ci. A (2) 8 (1960), 115-119.
      19. I. Pritsker, Polynomial inequalities, Mahler's measure, and multipliers, in Number Theory and Polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008, pp. 255-276.
      20. B. Szydlo, An application of some theorems of G. Szegö to Mahler measures of polynomials, Discuss. Math. 7 (1985), 145-148. Discuss. Math. 7 (1985), 145-148.
      21. J. D. Vaaler, An ABC inequality for Mahler's measure, Monatsh. Math. 154 (2008), no. 4, 323-343.

    8. Pisot and Salem Numbers.

      1. D. W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), 1244-1260.
      2. D. W. Boyd, Small Salem numbers, Duke Math. J. 44 (1976), 315-328.
      3. V. Flammang, M. Grandcolas, and G. Rhin, Small Salem numbers, in Number Theory in Progress, Vol. 1 (Zakopane-Ko\'scielisko, 1997), de Gruyter, Berlin, 1999, pp. 165-168.
      4. E. Ghate and E. Hironaka, The arithmetic and geometry of Salem numbers, Bull. Amer. Math. Soc. 38 (2001), 293-314.
      5. B. Gross, Unramified reciprocal polynomials and Coxeter decompositions, Mosc. Math. J. 2 (2002), no. 4, 681-692, 805.
      6. B. Gross and C. McMullen, Automorphisms of even unimodular lattices and unramified Salem numbers, J. Algebra 257 (2002), no. 2, 265-290.
      7. J. McKee and C. Smyth, Salem numbers, Pisot numbers, Mahler measure, and graphSalem numbers, Pisot numbers, Mahler measure, and graphss, Experiment. Math. 14 (2005), no. 2, 211-229.
      8. C. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math. 545 (2002), 201-233.
      9. C. McMullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. No. 95, (2002), 151-183.
      10. C. L. Siegel, Algebraic integers whose conjugates lie in the unit circle, Duke Math. J. 11 (1944), 597-602.

    9. Heights of Algebraic Numbers, Approximation Problems.

      1. F. Amoroso, Algebraic numbers close to 1 and variants of Mahler's measure, J. Number Theory 60 (1996), 80-96.
      2. F. Amoroso, Algebraic numbers close to 1: results and methods, pp. 305-316 in Number Theory (Tiruchirapalli, India 1996), ed. by V. K. Murty and M. Waldschmidt, Contemp. Math. 210, Amer. Math. Soc., Providence, 1998.
      3. Y. Bugeaud, Algebraic numbers close to 1 in non-Archimdean metrics, Ramanujan J. 2 (1998), 449-457.
      4. C. Doche, On the spectrum of the Zhang-Zagier height, Math. Comp. 70 (2001), 419-430.
      5. G. P. Dresden, Orbits of algebraic numbers with low heights, Math. Comp. 67 (1998), 815-820.
      6. G. P. Dresden, Sums of heights of algebraic numbers, Math. Comp. 72 (2003), no. 243, 1487-1499.
      7. A. Dubickas, On algebraic numbers of small measure, Lithuanian Math. J. 35 (1995), 333-342.
      8. A. Dubickas, On algebraic numbers close to 1, Bull. Austral. Math. Soc. 58 (1998), 423-434.
      9. A. Dubickas, Three problems for polynomials of small measure, Acta Arith. 98 (2001), 279-292.
      10. A. Dubickas, Mahler measures close to an integer, Canad. Math. Bull. 45 (2002), no. 2, 196-203.
      11. A. Dubickas and C. J. Smyth, On the Remak height, the Mahler measure and conjugate sets of algebraic numbers lying on two circles, Proc. Edinburgh. Math. Soc. 44 (2001), 1-17.
      12. M. Mignotte, Approximation des nombres algébriques par des nombres algébriques de grand degré, Ann. Fac. Sci. Toulouse Math. (6) 1 (1979), 165-170.
      13. M. Mignotte and M. Waldschmidt, On algebraic numbers of small height: linear forms in one logarithm, J. Number Theory 47 (1994), 43-62.
      14. D. Zagier, Algebraic numbers close to both 0 and 1, Math. Comp. 61 (1993), 485-491.

    10. Schinzel-Zassenhaus Conjecture.

      1. D. W. Boyd, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985), no. 171, 243-249.
      2. T. Callahan, M. Newman, and M. Sheingorn, Fields with large Kronecker constants, J. Number Theory 9 (1977), no. 2, 182-186.
      3. E. Dobrowolski, On the maximal modulus of conjugates of an algebraic integer, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 4, 291-292.
      4. A. Dubickas, On a conjecture of Schinzel and Zassenhaus, Acta Arith. 63 (1993), 15-20.
      5. A. Dubickas, The maximal conjugate of a non-reciprocal algebraic integer, Lithuanian Math. J. 37 (1997), no. 2, 129-133.
      6. A. Schinzel and H. Zassenhaus, A refinement of two theorems of Kronecker, Mich. Math. J. 12 (1965), 81-85.

    11. Applications of Mahler Measure, especially of Polynomials with Small Measure.

      1. D. H. Bailey and D. J. Broadhurst, A seventeenth-order polylogarithm ladder, 1999, arXiv:math.CA/9906134, 18 pp.
      2. F. Beaucoup, P. Borwein, D. Boyd, and C. Pinner, Multiple roots of [-1,1] power series, J. London Math. Soc. 57 (1998), no. 1, 135-147.
      3. H. Cohen, L. Lewin, and D. Zagier, A sixteenth-order polylogarithm ladder, Experiment. Math. 1 (1992), 25-34.
      4. W. Duke, A combinatorial problem related to Mahler's measure, Bull. Lond. Math. Soc. 39 (2007), no. 5, 741-748.
      5. M. Einsiedler, G. Everest, and T. Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.
      6. D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461-479.
      7. M. J. Mossinghoff, Polynomials with restricted coefficients and prescribed noncyclotomic factors, LMS J. Comput. Math. 6 (2003), 314-325.
      8. C. G. Pinner and J. D. Vaaler, The number of irreducible factors of a polynomial III, in Number Theory in Progress, Vol. 1 (Zakopane-Ko\'scielisko, 1997), de Gruyter, Berlin, 1999, pp. 395-405.
      9. J. H. Silverman, Exceptional units and numbers of small Mahler measure, Experiment. Math. 4 (1995), 69-83.
      10. J. H. Silverman, Small Salem numbers, exceptional units, and Lehmer's conjecture, Rocky Mountain J. Math. 26 (1996), 1099-1114.

  2. Mahler's Measure of Polynomials in Several Variables.

    1. General Theory and Inequalities.

      1. F. Amoroso, On the Mahler measure in several variables, Bull. Lond. Math. Soc. 40 (2008), no. 4, 619-630.
      2. F. Amoroso and M. Mignotte, Upper bounds for the coefficients of irreducible integer polynomials in several variables, Acta Arith. 99 (2001), no. 1, 1-12.
      3. F. Amoroso and U. Zannier, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 711-727.
      4. D. W. Boyd, Kronecker's theorem and Lehmer's problem for polynomials in several variables, J. Number Theory 13 (1981), 116-121.
      5. D. W. Boyd, Speculations concerning the range of Mahler's measure, Canad. Math. Bull. 24 (1981), 453-469.
      6. D. W. Boyd, Uniform approximation to Mahler's measure in several variables, Canad. Math. Bull. 41 (1998), no. 1, 125-128.
      7. W. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983), 356-362.
      8. P. Lelong, Mesure de Mahler et calcul de constantes universelles pour les polynômes de n variables, Math. Ann. 299 (1994), no. 4, 673-695.
      9. P. Lelong, Mesure de Mahler des polynômes et majoration par convexité, C. R. Acad. Sci Paris Sér. I Math. 315 (1992), no. 2, 139-142.
      10. K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962), 341-344.
      11. G. Myerson, A measure for polynomials in several variables, Canad. Math. Bull. 27 (1984), no. 2, 185-191.
      12. I. Z. Ruzsa, On Mahler's measure for polynomials in several variables, in Number Theory in Progress, Vol. 1 (Zakopane-Ko\'scielisko, 1997), de Gruyter, Berlin, 1999, pp. 431-444.
      13. A. Schinzel, On the Mahler measure of polynomials in many variables, Acta Arith. 79 (1997), 77-81.
      14. C. J. Smyth, A Kronecker-type theorem for complex polynomials in several variables, Canad. Math. Bull. 24 (1981), no. 4, 447-452. Addenda and errata, ibid. 25 (1982), no. 4, 504.

    2. Computations.

      1. D. W. Boyd and M. J. Mossinghoff, Small limit points of Mahler's measure, Experiment. Math. 14 (2005), no. 4., 403-413.
      2. G. Everest, Estimating Mahler's measure, Bull. Austral. Math. Soc. 51 (1995), no. 1, 145-151.

    3. Explicit Formulas and Identities; Connections with L-functions and Dilogarithms.

      1. M. J. Bertin, Mahler's measure: From number theory to geometry, in Number Theory and Polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008, pp. 20-32.
      2. M. Bertin, Une mesure de Mahler explicite, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 1, 1--3.
      3. M. Bertin, Mesure de Mahler d'une famille de pôlynomes, J. Reine Angew. Math. 569 (2004), 175-188.
      4. M. Bertin, Mesure de Mahler et régulateur elliptique: preuve de deux relations "exotiques", in Number Theory, CRM Proc. Lecture Notes 36, Amer. Math. Soc., Providence, RI, 2004, pp. 1-12.
      5. S. Boughzala, Mesure de Mahler et polylogarithmes, C. R. Math. Acad. Sci. Paris 338 (2004), no. 10, 747-750.
      6. S. Boughzala, Mesure de Mahler d'une famille de polynômes de deux variables, Publ. Math. Debrecen 68 (2006), no. 1-2, 25-36.
      7. D. W. Boyd, Mahler's measure and special values of L-functions, Experiment. Math., 37 (1998), 37-82.
      8. D. W. Boyd, Mahler's measure and special values of L-functions --- some conjectures, in Number Theory in Progress, Vol. 1 (Zakopane-Ko\'scielisko, 1997), de Gruyter, Berlin, 1999, pp. 27-34.
      9. D. W. Boyd and F. Rodriguez Villegas, Mahler's measure and the dilogarithm, I, Canad. J. Math. 54 (2002), no. 3, 468-492.
      10. J. Condon, Calculation of the Mahler measure of a three variable polynomial, preprint, 2003.
      11. C. D'Andrea and M. Lalín, On the Mahler measure of resultants in small dimensions, J. Pure Appl. Algebra 209 (2007), no. 2, 393-410.
      12. C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain Zn-actions, J. Amer. Math. Soc. 10 (1997), no. 2, 259-281.
      13. M. Lalín, Some examples of Mahler measures as multiple polylogarithms, J. Number Theory 103 (2003), no. 1, 85-108.
      14. M. Lalín, Mahler measure of some $n$-variable polynomial families, J. Number Theory 116 (2006), no. 1, 102-139.
      15. M. Lalín, On certain combination of colored multizeta values, J. Ramanujan Math. Soc. 21 (2006), no. 1, 115-127.
      16. M. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 138 (2007), no. 3, 391-422.
      17. M. Lalín, Functional equations for Mahler measures of genus-one curves, Algebra Number Theory 1 (2007), no. 1, 87-117.
      18. M. Lalín, Mahler measures and computations with regulators, J. Number Theory 128 (2008), no. 5, 1231-1271.
      19. M. Lalín, On a conjecture by Boyd, Int. J. Number Theory, to appear.
      20. H. Oyanagi, Differential equations for Mahler measures, J. Ramanujan Math. Soc. 18 (2003), no. 2, 181-194.
      21. G. A. Ray, Relations between Mahler's measure and values of L-series, Canad. J. Math. 39 (1987), 694-732.
      22. F. Rodriguez Villegas, Modular Mahler measures, I, in Topics in Number Theory (University Park, PA, 1997), Math. Appl. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 17-48.
      23. F. Rodriguez Villegas, Identities between Mahler measures, in Number Theory for the Millennium, III (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 223-229.
      24. F. Rodriguez Villegas, R. Toledano, and J. D. Vaaler, Estimates for Mahler's measure of a linear form, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 2, 473-494.
      25. M. Rogers, A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities, J. Number Theory 121 (2006), no. 2, 265-304.
      26. C. J. Smyth, On measures of polynomials in several variables, Bull. Austral. Math. Soc. 23 (1981), no. 1, 49-63. Corrigendum, C. J. Smyth and G. Myerson, ibid. 26 (1982), no. 2, 317-319.
      27. C. J. Smyth, An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Théor. Nombres Bordeaux 14 (2002), no. 2, 683-700.
      28. C. J. Smyth, Explicit formulas for the Mahler measure of families of multivariable polynomials, preprint.
      29. N. Touafek, Mahler's measure: proof of two conjectured formulae, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 16 (2008), no. 2, 127-136.
      30. N. Touafek and M. Kerada, Mahler measure and elliptic regulator: some identities, JP J. Algebra Number Theory Appl. 8 (2007), no. 2, 271-285.
      31. S. Vandervelde, A formula for the Mahler measure of axy+bx+cy+d, J. Number Theory 100 (2003), no. 1, 184-202.
      32. S. Vandervelde, The Mahler measure of parametrizable polynomials, J. Number Theory 128 (2008), no. 8, 2231-2250.

  3. Generalizations of Mahler's Measure and Analogues of Lehmer's Problem: Higher Dimension, Elliptic Mahler Measure, Elliptic Curves, Abelian Varieties.

    1. F. Amoroso and S. David, Le théorème de Dobrowolski en dimension supérieure, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 10, 1163-1166.
    2. F. Amoroso and S. David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179.
    3. F. Amoroso and S. David, Minoration de la hauteur normalisée des hypersurfaces, Acta Arith. 92 (2000), no. 4, 339-366.
    4. M. Anderson and D. Masser, Lower bound for heights on elliptic curves, Math. Zeit. 174 (1980), 23-34.
    5. M. Baker, Lower bounds for the canonical height on elliptic curves over abelian extensions, Int. Math. Res. Not. 2003, no. 29, 1571-1589.
    6. M. Bertin, Mesure de Mahler et régulateur: preuve de deux relations "exotiques", in Number Theory, CRM Proc. Lecture Notes 36, Amer. Math. Soc., 2004, pp. 1-12.
    7. M. Bertin, Mahler's measure and L-series of K3 hypersurfaces, in Mirror Symmetry V, AMS/IP Stud. Adv. Math. 38, Amer. Math. Soc., Providence, RI, 2006, pp. 3-18.
    8. M. Bertin, Mesure de Mahler d'hypersurfaces K3, J. Number Theory 128 (2008), no. 11, 2890-2913.
    9. A. Besser and C. Deninger, p-adic Mahler measures, J. Reine Angew. Math. 517 (1999), 19-50.
    10. M. Carrizosa, Problème de Lehmer et variéétés abéliennes CM, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23-24, 1219-1224.
    11. S. David and M. Hindry, Minoration de la hauteur de Néron-Tate sur les variétés abéliennes de type C.M., J. Reine Angew. Math. 529 (2000), 1-74.
    12. S. David and A. Pacheco, Le problème de Lehmer abélien pour un module de Drinfelʹd, Int. J. Number Theory 4 (2008), no. 6, 1043-1067.
    13. G. Everest, On the elliptic analogue of Jensen's formula, J. London Math. Soc. (2) 59 (1999), no. 1, 21-36.
    14. G. Everest and B. N. Fhlathúin, The elliptic Mahler measure, Math. Proc. Cambridge Philos. Soc. 120 (1996), 13-25.
    15. G. Everest and C. Pinner, Bounding the elliptic Mahler measure II, J. London Math. Soc. (2) 58 (1998), 1-8. Corrigendum, ibid. 62 (2000), no. 2, 640.
    16. Y. Gon and H. Oyanagi, Generalized Mahler measures and multiple sine functions, Internat. J. Math. 15 (2004), no. 5, 425-442.
    17. M. Hindry and J. H. Silverman, On Lehmer's conjecture for elliptic curves, in Séminaire de Théorie des Nombres, Paris (1988-1989), Progr. Math. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 103-116.
    18. N. Kurokawa, A q-Mahler measure, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 5, 70-73.
    19. N. Kurokawa, M. Lalín, and H. Ochiai, Higher Mahler measures and zeta functions, Acta Arith. 135 (2008), no. 3, 269-297.
    20. N. Kurokawa and H Ochiai, Mahler measures via the crystalization, Comment. Math. Univ. St. Pauli 54 (2005), no. 2, 121-137.
    21. M. Laurent, Minoration de la hauteur de Néron-Tate, in Séminaire de Théorie de Nombres, Paris (1981-1982) 38 (1983), Birkhäuser, Boston-Basel-Stuttgart, 137-152.
    22. D. Lind, Lehmer's problem for compact abelian groups, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1411-1416.
    23. D. Masser, Small values of the quadratic part of the Néron-Tate height on an abelian variety, Compositio Math. 53 (1984), 153-170.
    24. D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117 (1989), 247-265.
    25. H. Oyanagi, q-analogues of Mahler measures, J. Ramanujan Math. Soc. 19 (2004), no. 3, 203-212.
    26. A. Pacheco, Lehmer's conjecture in positive characteristic, preprint, April 2002.
    27. C. Pinner, Bounding the elliptic Mahler measure, Math. Proc. Cambridge Philos. Soc. 124 (1998), 521-529.
    28. C. Pontreau, Minoration effective de la hauteur des points d'une courbe de Gm2 sur Q, Univ. Caen Rapport de recherche 18 (2004).
    29. N. Ratazzi, Problème de Lehmer pour les hypersurfaces de variétés abéliennes de type C.M., Acta Arith. 113 (2004), no. 3, 273-290.
    30. J. H. Silverman, Lower bound for the canonical height on elliptic curves, Duke Math. J. 48 (1981), 633-648.
    31. J. H. Silverman, The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), 723-743.
    32. S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), 159-165.

  4. Dynamical Systems.

    1. P. D'Ambros, G. Everest, R. Miles, and T. Ward, Dynamical systems arising from elliptic curves, Colloq. Math. 84/85 (2000), part 1, 95-107.
    2. M. Einsiedler, A generalisation of Mahler measure and its application in algebraic dynamical systems, Acta Arith. 88 (1999), no. 1, 15-29.
    3. D. A. Lind, Ergodic automorphisms of the infinite torus are Bernoulli, Israel J. Math. 17 (1974), 162-168.
    4. D. Lind, K. Schmidt, and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), 593-629.
    5. P. Moussa, Ensembles de Julia et propriétés de localisation des entiers algébriques, exp. no. 21 in Seminar on Number Theory, 1984-1985 (Talence, 1984/1985), Univ. Bordeaux I, 1985, 10 pp.
    6. P. Moussa, Diophantine properties of Julia sets, in Chaotic Dynamics and Fractals (Atlanta, Ga., 1985), Academic Press, 1986, pp. 215-227.
    7. P. Moussa, J. S. Geronimo, and D. Bessis, Ensembles de Julia et propriétés de localisation des familles itérées d'entiers algébriques, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 8, 281-284.
    8. J. Pineiro, L. Szpiro, and T. J. Tucker, Mahler measure for dynamical systems on P1 and intersection theory on a singular arithmetic surface, in Geometric Methods in Algebra and Number Theory, Progr. Math. 235, Birkhäuser, Boston, MA, 2005, pp. 219-250.

  5. Geometry and Topology.

    1. D. W. Boyd, Mahler's measure and invariants of hyperbolic manifolds, in Number Theory for the Millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 127-143.
    2. A. Champanerkar and I. Kofman, On the Mahler measure of Jones polynomials under twisting, Algebr. Geom. Topol. 5 (2005), 1-22.
    3. E. Hironaka, The Lehmer polynomial and pretzel knots, Canad. Math. Bull. 44 (2001), no. 4, 440-451.
    4. E. Hironaka, Lehmer's problem, McKay's correspondence, and 2,3,7, in Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis, MD, 2001), Contemp. Math. 324, Amer. Math. Soc., Providence, RI, 2003, pp. 123-138.
    5. E. Hironaka, On hyperbolic perturbations of algebraic links and small Mahler measure, in Singularities in Geometry and Topology 2004, Adv. Stud. Pure Math. 46, Math. Soc. Japan, Tokyo, 2007, pp. 77--94.
    6. M. Lalín, Mahler measure and volumes in hyperbolic space, Geom. Dedicata 107 (2004), 211-234.
    7. C. J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol. 8 (2004), 1301-1359.
    8. C. McMullen, Coxeter groups, Salem numbers and the Hilbert metric, Publ. Math. Inst. Hautes Études Sci. No. 95, (2002), 151-183.
    9. H. Murakami, Mahler measure of the colored Jones polynomial and the volume conjecture, arXiv:math.GT/0206249, June 2002.
    10. R. Riley, Growth of order of homology of cyclic branched covers of knots, Bull. London Math. Soc. 22 (1990), 287-297.
    11. D. S. Silver, A. Stoimenow, and S. G. Williams, Euclidean Mahler measure and twisted links, Algebr. Geom. Topol. 6 (2006), 581--602.
    12. D. S. Silver and S. G. Williams, Mahler measure of Alexander polynomials, J. London Math. Soc. (2) 69 (2004), no. 3, 767-782.
    13. D. S. Silver and S. G. Williams, Coloring link diagrams with a continuous palette, Topology, 39 (2000), 1225-1237.
    14. D. S. Silver and S. G. Williams, Mahler measure, links and homology growth, Topology 41 (2002), no. 5, 979-991.
    15. D. S. Silver and S. G. Williams, Torsion numbers of augmented groups with applications to knots and links, Enseign. Math. (2) 48 (2002), no. 3-4, 317-343.

  6. Surveys.

    1. D. W. Boyd, Variations on a theme of Kronecker, Canad. Math. Bull. 21 (1978), 129-133.
    2. G. Everest, Measuring the height of a polynomial, Math. Intelligencer 20 (1998), no. 3, 9-16.
    3. E. Ghate and E. Hironaka, The arithmetic and geometry of Salem numbers, Bull. Amer. Math. Soc. 38 (2001), 293-314.
    4. E. Hironaka, What is Lehmer's number?, Notices Amer. Math. Soc. 56 (2009), no. 3, 374-375.
    5. J. Hunter, Algebraic integers on the unit circle, Math. Chronicle 11 (1982), no. 1-2, 37-47.
    6. A. Schinzel, The Mahler measure of polynomials, in Number Theory and its Applications, Lec. Notes in Pure and Appl. Math. 204, Marcel-Dekker, New York, 1999, pp. 171-183.
    7. S. Siciliano, Lehmer's conjecture, Pure Math. Appl. 11 (2000), no. 3, 539-541.
    8. M. Waldschmidt, Sur le produit des conjugués extérieurs au cercle unité d'un entier algébrique, Enseign. Math. (2) 26 (1980), no. 3-4, 201-209.
    9. C. Smyth, The Mahler measure of algebraic numbers: A survey, in Number Theory and Polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008, pp. 322-349.

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