Integer transfinite diameter and polynomials with small Mahler measure

Flammang, Valérie; Rhin, Georges; Sac-Épée, Jean-Marc

doi:10.1090/S0025-5718-06-01791-1

Integer transfinite diameter and polynomials with small Mahler measure
HTML articles powered by AMS MathViewer

by Valérie Flammang, Georges Rhin and Jean-Marc Sac-Épée PDF

Math. Comp. 75 (2006), 1527-1540 Request permission

Abstract:

In this work, we show how suitable generalizations of the integer transfinite diameter of some compact sets in $\mathbb {C}$ give very good bounds for coefficients of polynomials with small Mahler measure. By this way, we give the list of all monic irreducible primitive polynomials of $\mathbb {Z}[X]$ of degree at most $36$ with Mahler measure less than $1. 324...$ and of degree $38$ and $40$ with Mahler measure less than $1. 31$.

References

Peter Borwein, Computational excursions in analysis and number theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 10, Springer-Verlag, New York, 2002. MR 1912495, DOI 10.1007/978-0-387-21652-2
Peter Borwein and Tamás Erdélyi, The integer Chebyshev problem, Math. Comp. 65 (1996), no. 214, 661–681. MR 1333305, DOI 10.1090/S0025-5718-96-00702-8
David W. Boyd, Reciprocal polynomials having small measure, Math. Comp. 35 (1980), no. 152, 1361–1377. MR 583514, DOI 10.1090/S0025-5718-1980-0583514-9
David W. Boyd, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985), no. 171, 243–249, S17–S20. MR 790657, DOI 10.1090/S0025-5718-1985-0790657-8
David W. Boyd, Reciprocal polynomials having small measure. II, Math. Comp. 53 (1989), no. 187, 355–357, S1–S5. MR 968149, DOI 10.1090/S0025-5718-1989-0968149-6
D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461–479. MR 1503118, DOI 10.2307/1968172
V. Flammang and G. Rhin, Algebraic integers whose conjugates all lie in an ellipse, Math. Comp. 74 (2005), 2007–2015.
V. Flammang, G. Rhin, and C. J. Smyth, The integer transfinite diameter of intervals and totally real algebraic integers, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 137–168 (English, with English and French summaries). MR 1469665
V. Flammang, M. Grandcolas, and G. Rhin, Small Salem numbers, Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 165–168. MR 1689505
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
Michael J. Mossinghoff, Polynomials with small Mahler measure, Math. Comp. 67 (1998), no. 224, 1697–1705, S11–S14. MR 1604391, DOI 10.1090/S0025-5718-98-01006-0
C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, GP-Pari version 2.0.12, 1998.
I. Pritsker, Small polynomials with integer coefficients (preprint).
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. MR 2043995
C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169–175. MR 289451, DOI 10.1112/blms/3.2.169
C. J. Smyth, The mean values of totally real algebraic integers, Math. Comp. 42 (1984), no. 166, 663–681. MR 736460, DOI 10.1090/S0025-5718-1984-0736460-5
C.J. Smyth, Private communication, 2003.
Paul Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), no. 1, 81–95. MR 1367580, DOI 10.4064/aa-74-1-81-95
http://www.math.univ-metz.fr/~rhin.
http://www.cecm.sfu.ca/~mjm/Lehmer/.
Qiang Wu, On the linear independence measure of logarithms of rational numbers, Math. Comp. 72 (2003), no. 242, 901–911. MR 1954974, DOI 10.1090/S0025-5718-02-01442-4