Open Access

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions


Cite

[AM] ADLER, R. L.—MARCUS, B.: Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc. 20 (1979), no. 219, iv–84.Search in Google Scholar

[A1] AMOROSO, F.: Sur des polynômes de petites mesures de Mahler, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 11-14.Search in Google Scholar

[A2] AMOROSO, F.: Algebraic numbers close to 1: results and methods, in: Number Theory (Tiruchirapalli, India 1996), (V. K. Murty and M. Waldschmidt, Eds.) Amer. Math. Soc., Providence, Contemp. Math. 210 (1998), pp. 305–316.Search in Google Scholar

[ADd1] AMOROSO, F.—DAVID, S.: Le théorème de Dobrowolski en dimension supérieure, C. R. Acad. Sci. paris Sér. I Math. 326 (1998), 1163–1166.10.1016/S0764-4442(98)80219-XSearch in Google Scholar

[ADd2] AMOROSO, F.—DAVID, S.: Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145–179.10.1515/crll.1999.058Search in Google Scholar

[ADn] AMOROSO, F.—DELSINNE, S.: Une minoration relative explicite pour la hauteur dans une extension d’une extension abélienne, Diophantine geometry, CRM Series 4, ed. Norm., Pisa (2007), 1–24.Search in Google Scholar

[AD] AMOROSO, F.—DVORNICICH, R.: A lower bound for the height in abelian extensions, J. Number Theory 80 (2000), 260–272.10.1006/jnth.1999.2451Open DOISearch in Google Scholar

[AM] AMOROSO, F.—MIGNOTTE, M.: On the distribution of the roots of polynomials, Ann. Inst. Fourier 46 (1996), 1275–1291.10.5802/aif.1548Open DOISearch in Google Scholar

[AZ1] AMOROSO, F.—ZANNIER, U.: A lower bound for the height in Abelian extensions, J. Number Theory 80 (2000), 260–272.10.1006/jnth.1999.2451Open DOISearch in Google Scholar

[AZ2] AMOROSO, F.—ZANNIER, U.: A uniform relative Dobrowolski’s lower bound over abelian extensions, Bull. London Math. Soc. 42 (2010), 489–498.10.1112/blms/bdq008Open DOISearch in Google Scholar

[Al] APOSTOL, T. M.: Zeta and related functions, NIST handbook of mathematical functions, (F. W. F. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, Eds.) National Institute of Standards and Technology, Washington, DC, and Cambridge University Press, Cambridge, 2010, 601–616.Search in Google Scholar

[Bk] BAKER, M.: Canonical heights on elliptic curves over abelian extensions, Int.Math. Res. Not. 29 (2003), 1571–1589.10.1155/S1073792803212083Open DOISearch in Google Scholar

[B-S] BERTIN, M. J.—DECOMPS-GUILLOUX, A.—GRANDET-HUGOT, M.—PATHIAUX-DELEFOSSE, M.—SCHREIBER, J. P.: Pisot and Salem Numbers, (with a preface by David W. Boyd.), Birkhaüser Verlag, Basel 1992.10.1007/978-3-0348-8632-1Search in Google Scholar

[Bn] BERTIN, M. J.: Quelques résultats nouveaux sur les nombres de Pisot et de Salem, Number Theory in Progress, Vol. I, An Intern. Conf. on Number Theory, org. Stefan Banach Int. Math. Research Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, 1997, (K. Győry, H. Iwaniec, J. Urnanowicz, W., Eds.) de Gruyter, Berlin (1999),, pp. 1–10.Search in Google Scholar

[BD] BESSER, A.—DENINGER, C.: p-adic Mahler measures, J. Reine Angew.Math., 517 (1999), 19–50.10.1515/crll.1999.093Search in Google Scholar

[Bu] BILU, Y.: Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), 465–476.10.1215/S0012-7094-97-08921-3Open DOISearch in Google Scholar

[ByM] BLANSKY, P. E.—MONTGOMERY, H. L.: Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369.10.4064/aa-18-1-355-369Search in Google Scholar

[Bl] BOREL, E.: Leçons sur les Séries Divergentes, Gauthier-Villars, 2e édition, Paris, 1928.Search in Google Scholar

[BDM] BORWEIN, P.—DOBROWOLSKI, E.—MOSSINGHOFF, M. J.: Lehmer’s problem for polynomials with odd coefficients, Ann. of Math. 166 (2007), 347–366.10.4007/annals.2007.166.347Search in Google Scholar

[BS] BORWEIN, P.—STRAUB, A.: Mahler measures, short walks and log-sine integrals, Theoret. Comput. Sci. 479 (2013), 4–21.10.1016/j.tcs.2012.10.025Search in Google Scholar

[Bo0] BOYD, D. W.: Pisot numbers and the width of meromorphic functions, privately circulated manuscript (January 1977).Search in Google Scholar

[Bo1] BOYD, D. W.: Variations on a Theme of Kronecker, Canad. Math. Bull. 21 (1978), 129–133.10.4153/CMB-1978-023-xOpen DOISearch in Google Scholar

[Bo2] BOYD, D. W.: Kronecker’s Theorem and Lehmer’s Problem for polynomials in Several Variables, J. Number Th. 13 (1981), 116–121.10.1016/0022-314X(81)90033-0Search in Google Scholar

[Bo3] BOYD, D. W.: Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), 453–469.10.4153/CMB-1981-069-5Open DOISearch in Google Scholar

[Bo4] BOYD, D. W.: The maximal modulus of an algebraic integer, Math. Comp. 45 (1985), 243–249.10.1090/S0025-5718-1985-0790657-8Open DOISearch in Google Scholar

[BM] BOYD, D. W.—MOSSINGHOFF, M. J.: Small Limit Points of Mahler’s Measure, Exp. Math. 14 (2005), 403–414.10.1080/10586458.2005.10128936Search in Google Scholar

[Br] BREUSCH, R.: On the distribution of the roots of a polynomial with integral coefficients, Proc. Amer. Math. Soc. 2 (1951), 939–941.10.1090/S0002-9939-1951-0045246-9Open DOISearch in Google Scholar

[CS] CANTOR, D. C.—STRAUSS, E. G.: On a conjecture of D.H Lehmer, Acta Arith. 42 (1982/83), 97–100. Correction: ibid. 42 (3) (1983), 327.10.4064/aa-42-1-97-100Search in Google Scholar

[Ca] CASSELS, J.W.S.: On a problem of Schinzel and Zassenhaus, J. Math. Sciences 1 (1966), 1–8.Search in Google Scholar

[CV] CHERN, S.-J.—VAALER, J. D.: The distribution of values of Mahler’s measure, J. Reine Angew. Math. 540 (2001), 1–47.10.1515/crll.2001.084Search in Google Scholar

[C] COPSON, E. T.: Asymptotic Expansions, (Reprint of the 1965 original), in: Cambridge Tracts in Math., Vol. 55, Cambridge University Press, Cambridge, 2004.Search in Google Scholar

[DH] DAVID, S.—HINDRY, M.: Minoration de la hauteur de Néron-Tate sur les variétés de type C.M., J. Reine Angew. Math. 529 (2000), 1–74.10.1515/crll.2000.096Search in Google Scholar

[Di] DINGLE, R. B.: Asymptotic Expansions: their Derivation and Interpretation, Academic Press, London-New York, 1973.Search in Google Scholar

[DDs] DIXON, J. D.—DUBICKAS, A.: The values of Mahler Measures, Mathematika 51 (2004), 131–148.10.1112/S0025579300015564Open DOISearch in Google Scholar

[Do1] DOBROWOLSKI, E.: On the Maximal Modulus of Conjugates of an Algebraic Integer, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 291–292.Search in Google Scholar

[Do2] DOBROWOLSKI, E.: On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391–401.10.4064/aa-34-4-391-401Search in Google Scholar

[De] DOCHE, C.: Zhang-Zagier heights of perturbed polynomials, J. Théor. Nombres Bordeaux 13 (2001), 103–110.10.5802/jtnb.307Search in Google Scholar

[Ds1] DUBICKAS, A.: On a conjecture of A. Schinzel and H. Zassenhaus, Acta Arith. 63 (1993), 15–20.10.4064/aa-63-1-15-20Search in Google Scholar

[Ds2] DUBICKAS, A.: On algebraic numbers of small measure, Lithuanian Math. J. 35 (1995/1996), 333–342.10.1007/BF02348822Open DOISearch in Google Scholar

[Ds3] DUBICKAS, A.: The maximal conjugate of a non-reciprocal algebraic integer, Lithuanian Math. J. 37 (2) (1997/1998), 129–133.10.1007/BF02465885Open DOISearch in Google Scholar

[Ds4] DUBICKAS, A.: Nonreciprocal algebraic numbers of small measure, Comment. Math. Univ. Carolin. 45 (2004), 693–697.Search in Google Scholar

[Ds5] DUBICKAS, A.: On numbers which are Mahler measures, Monatsh. Math. 141 (2004), 119–126.10.1007/s00605-003-0010-0Search in Google Scholar

[E] ERDÉLYI, A.: Asymptotic Expansions, Dover Publications, New York 1956.10.21236/AD0055660Search in Google Scholar

[ET] ERDÖS, P.—TURÁN, P.: On the distribution of roots of polynomials, Ann. Math. 51 (1950), 105–119.10.2307/1969500Search in Google Scholar

[F] FLAMMANG, V.: The Mahler measure of trinomials of height 1, J. Aust.Math. Soc. 96 (2014), 231–243.10.1017/S1446788713000633Open DOISearch in Google Scholar

[FGR] FLAMMANG, V.—GRANDCOLAS, M.—RHIN, G.: Small Salem numbers, in: Number Theory in Progress, Vol. 1, (Zakopane-Kościelisko, 1997), de Gruyter, Berlin (1999), 165–168.10.1515/9783110285581.165Search in Google Scholar

[FLP] L. FLATTO, L.—LAGARIAS, J. C.—POONEN, B.: The zeta function of the beta-transformation, Ergod. Th. Dynam. Sys. 14 (1994), 237–266.10.1017/S0143385700007860Search in Google Scholar

[GM] GALATEAU, A.—MAHÉ, V.: Some consequences of Masser’s counting Theorem on Elliptic Curves, (2015).Search in Google Scholar

[G] GANELIUS, T.: Sequences of analytic functions and their zeros, ArkivMath. 3 (1953), 1–50.Search in Google Scholar

[HS] HINDRY, M.—SILVERMAN, J.: On Lehmer’s conjecture for elliptic curves, in: Séminaire de Théorie des Nombres, Paris 1988/1989, Progress in Math. Vol. 91, Birkhäuser, Paris (1990), pp. 103–116.Search in Google Scholar

[Lg] LANGEVIN, M.: Calculs explicites de constantes de Lehmer, in: Groupe de Travail en Théorie Analytique et Élémentaire des Nombres, 1986/1987, Publ. Math. Orsay, Univ. Paris XI, Orsay, Vol. 88, 1988, pp. 52–68.Search in Google Scholar

[La] LAURENT, M.: Minoration de la hauteur de Néron-Tate, in: Séminaire de Théorie des Nombres Paris 1981/1982, Progress in Math., Vol. 38, Birkhäuser, Paris, 1983, pp. 137–152.Search in Google Scholar

[La2] LAURENT, M.: Sur quelques résultats récents de transcendance, Some recent results concerning transcendence, Astérisque, Journées Arithmétiques1989 (Luminy 1991/1992), 209–230.Search in Google Scholar

[Lw] LAWTON, W. M.: A Problem of Boyd Concerning Geometric Means of Polynomials, J. Number Theory 16 (1983), 356–362.10.1016/0022-314X(83)90063-XOpen DOISearch in Google Scholar

[Le] LEHMER, D. H.: Factorization of certain cyclotomic functions, Ann. Math. 34 (1933), 461–479.10.2307/1968172Search in Google Scholar

[Ln] LEWIN, L.: Polylogarithms and Associated Functions, with a foreword by A. J. Van der Poorten. North-Holland Publishing Co., New York-Amsterdam, 1981.Search in Google Scholar

[Lt] LOUBOUTIN, R.: Sur la mesure de Mahler d’un nombre algébrique, C. R. Acad. Sci. Paris Série I, t. 296 (1983), 707–708.Search in Google Scholar

[Ma] MASSER, D. W.: Counting points of small height on elliptic curves Bull. Soc. Math. France 117 (1989), 247–265.,Search in Google Scholar

[Mv] MATVEEV, E.M.: On the cardinality of algebraic integers, Math. Notes 49 (1991), 437–438.10.1007/BF01158227Open DOISearch in Google Scholar

[Me] MEYER, M.: Le problème de Lehmer, méthode de Dobrowolski et lemme de Siegel “à la Bombieri-Vaaler”, Publ. Math. Univ. P. et M. Curie (Paris VI), 90, Problèmes Diophantiens, 1988/1989, No 5.Search in Google Scholar

[Mt0] MIGNOTTE, M.: 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, Paris (1978).Search in Google Scholar

[Mt1] MIGNOTTE, M.: Sur un théorème de M. Langevin, Acta Arith. 54 (1989), 81–86.10.4064/aa-54-1-81-86Search in Google Scholar

[Mt2] MIGNOTTE, M.: Remarque sur une question relative à des fonctions conjuguées, C. R. Acad. Sci. Paris, Série I, t. 315 (1992), 907–911.Search in Google Scholar

[Mf] MOSSINGHOFF, M. J.: Polynomials with small Mahler measure, Math. Comp. 67 (1998), 1697–1705, S11-S14.10.1090/S0025-5718-98-01006-0Open DOISearch in Google Scholar

[MfL] MOSSINGHOFF, M. J.: Known polynomials through degree 180, http://www.cecm.sfu.caz/~mjm/Lehmer, (1996); implemented (2001): P. Lisonek; and (2003): G. Rhin and J.-M. Sac-Epée; complete through degree 40.Search in Google Scholar

[MRW] MOSSINGHOFF, M. J.—RHIN, G.—WU, Q.: Minimal Mahler Measures, Experimental Math. 17 (2008), 451–458.10.1080/10586458.2008.10128872Open DOISearch in Google Scholar

[Pe] PETSCHE, C.: A quantitative version of Bilu’s equidistribution theorem, Int. J. Number Theory 1 (2005), 281–291.10.1142/S1793042105000145Open DOISearch in Google Scholar

[P] POINCARÉ, H.: Lȩcons de Mécanique Céleste Paris, Gauthier-Villars, t. I 1905, t. II-1 1907, t. II-2 1909, t. III 1910.Search in Google Scholar

[Pr] PRITSKER, I. E.: Distribution of algebraic numbers, J. Reine Agew. Math. 657 (2011), 5780.10.1515/crelle.2011.049Search in Google Scholar

[Rz] RATAZZI, N.: Théorème de Dobrowolski-Laurent pour les extensions abéliennes sur une courbe elliptique à multiplications complexes, Int. Math. Res. Not. 58 (2004), 3121–3152.10.1155/S1073792804140518Open DOISearch in Google Scholar

[Ra] RAUSCH, U.: On a theorem of Dobrowolski about the product of conjugate numbers, Colloq. Math. 50 (1985), 137–142.10.4064/cm-50-1-137-142Search in Google Scholar

[Rd] RÉMOND, G.: Intersection de sous-groupes et de sous-variétés I., Math. Ann. 333 (2005), 525-548.10.1007/s00208-005-0673-zSearch in Google Scholar

[Re] RÉNYI, A.: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.10.1007/BF02020331Open DOISearch in Google Scholar

[RS] RHIN, G.—SMYTH, C. J.: On the absolute Mahler measure of polynomials having all zeros in a sector, Math. Comp. 64 (1995), 295–304.10.1090/S0025-5718-1995-1257579-6Open DOISearch in Google Scholar

[RW] RHIN, G.—WU, Q.: On the absolute Mahler measure of polynomials having all zeros in a sector II, Math. Comp. 74 (2005), 383–388.10.1090/S0025-5718-04-01676-XOpen DOISearch in Google Scholar

[Sc1] SCHINZEL, A.: Reducibility of lacunary polynomials, Acta Arith. 16 (1969), 123–159.10.4064/aa-16-2-123-160Search in Google Scholar

[Sc2] SCHINZEL, A.: On the product of the conjugates outside the unit circle of an algebraic number, Acta Arith. 24 (1973), 385–399; Addendum: ibid. 26 (1974/75), 329–331.Search in Google Scholar

[Sc3] SCHINZEL, A.: On the Mahler measure of polynomials in many variables, Acta Arith. 79 (1997), 77–81.10.4064/aa-79-1-77-81Search in Google Scholar

[SZ] SCHINZEL, A.—ZASSENHAUS, H.: A refinement of two theorems of Kronecker, Michigan Math. J. 12 (1965), 81–85.10.1307/mmj/1028999247Open DOISearch in Google Scholar

[Sr] SELMER, E. S.: On the irreducibility of certain trinomials, Math. Scand. 4 (1956), 287–302.10.7146/math.scand.a-10478Open DOISearch in Google Scholar

[Sn] SILVERMAN, J. H.: Lehmer’s Conjecture for Polynomials Satisfying a Congruence Divisibility Condition and an Analogue for Elliptic Curves, J. Théorie Nombres Bordeaux 24 (2012), 751–772.10.5802/jtnb.820Search in Google Scholar

[Si] SINCLAIR, C.: The distribution of Mahler’s measures of reciprocal polynomials, Int. J. Math. Math. Sci. 49–52 (2004), 2773–2786.10.1155/S0161171204312469Search in Google Scholar

[Sy1] SMYTH, C.: On the product of the conjugates outside the unit circle of an algebraic integer, Bull. Lond. Math. Soc. 3 (1971), 169–175.10.1112/blms/3.2.169Open DOISearch in Google Scholar

[Sy2] SMYTH, C.: On measures of polynomials in several variables, Bull. Austral. Math. Soc. 23 (1981), 49–63.10.1017/S0004972700006894Open DOISearch in Google Scholar

[Sy3] SMYTH, C.: The Mahler measure of algebraic numbers: A Survey, in: Number Theory and Polynomials, London Math. Soc. Lecture Note Ser. Vol. 352, Cambridge Univ. press, Cambridge, 2008, pp. 322–349.10.1017/CBO9780511721274.021Search in Google Scholar

[Sy4] SMYTH, C.: Topics in the Theory of Numbers, PhD Thesis, Cambridge 1972.Search in Google Scholar

[Sff] STEFFENSEN, J. F.: Interpolation, (1927); reprint, 2nd ed. Chelsea Publ. Co, New York 1950.Search in Google Scholar

[St] STEWART, C. L.: Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. Math. France 106 (1978), 169–176.10.24033/bsmf.1868Search in Google Scholar

[SB] STOER, J.—BULIRSCH, R.: , Introduction to Numerical Analysis, in: Texts in Appl. Math. Vol. 12, 2nd ed., Springer-Verlag, New York, 1993.10.1007/978-1-4757-2272-7Search in Google Scholar

[VG] VERGER-GAUGRY, J.-L.: Uniform distribution of the Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers, Unif. Distrib. Theory 3 (2008), 157–190.Search in Google Scholar

[V] VOUTIER, P. M.: An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), 81–95.10.4064/aa-74-1-81-95Search in Google Scholar

[W0] WALDSCHMIDT, M.: Sur le produit des conjugués extérieurs au cercle, L’Enseign. Math. 26 (1980), 201–209.Search in Google Scholar

[W1] WALDSCHMIDT, M.: Auxiliary functions in transcendental number theory, SASTRA Ramanujan Lectures, Ramanujan J. 20 no. 3, (2009), 341–373.10.1007/s11139-009-9204-yOpen DOISearch in Google Scholar

[W2] WALDSCHMIDT, M.: Diophantine Approximation on Luinear Algebraic Group: Transcendence Properties of the Exponential Function in Several Variables, Grund. Math. Wiss., Vol. 326, Springer-Verlag, Berlin, 2000.Search in Google Scholar

[Wu] WU, Q.: The smallest Perron numbers, Math. Comp. 79 (2010), 2387–2394.10.1090/S0025-5718-10-02345-8Open DOISearch in Google Scholar

[Za] ZAGIER, D.: Algebraic numbers close to 0 and 1, Math. Comp. 61 (1993), 485–491.10.1090/S0025-5718-1993-1197513-9Search in Google Scholar

[Zi] ZAÏMI, T.: Sur les K-nombres de Pisot de petite mesure, Acta Arith. 77 (1996), 103–131.10.4064/aa-77-2-103-131Search in Google Scholar

eISSN:
2309-5377
Language:
English