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Speculations Concerning the Range of Mahler's Measure

Published online by Cambridge University Press:  20 November 2018

David W. Boyd*
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, B.C., Canada V6T 1Y4
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I would like to express my thanks to the Canadian Mathematical Society for inviting me to present this lecture. I would also like to express my appreciation to C.J. Smyth for numerous helpful conversations during his visit this year at the University of British Columbia.

This paper follows reasonably closely the outline of the lecture presented in Ottawa. More details are given here though and a number of proofs which would not be otherwise accessible have been added as Appendices. The attentive reader will soon realize the appropriateness of the title.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Apéry, R., VIrrationalité de Certaines Constantes de l'analyse, Les Journées arithmétiques de Marseille, Astérisqu. 61 (1979),Google Scholar
2. Blanksky, P. E. and Montgomery, H. L., Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355-369.Google Scholar
3. Boyd, D. W., Pisot sequences which satisfy no linear recurrence, Acta Arith. 32 (1977), 89-98.Google Scholar
4. Boyd, D. W., Small Salem numbers, Duke Math. Jour. 44 (1977), 315-328.Google Scholar
5. Boyd, D. W., Pisot numbers and the width of meromorphic functions, privately circulated manuscript, January 1977.Google Scholar
6. Boyd, D. W., Pisot and Salem numbers in Intervals of the Real Line, Math, of Comp. 32 (1978), 1244-1260.Google Scholar
7. Boyd, D. W., The successive derived sets of the Pisot numbers, Proc. Amer. Math. Soc. 73 (1979), 154-156.Google Scholar
8. Boyd, D. W., Variations on a theme of Kronecker, Canad. Math. Bull. 21 (1978), 129-133.Google Scholar
9. Boyd, D. W., Kronecker's theorem and Lehmer's problem for polynomials in several variables, Jour. Number Thry. 12, 1980.Google Scholar
10. Clarke, B., Asymptotes and Intercepts of Real-Power Polynomial Surfaces from the Geometry of the Exponent Polytope, SIAM J. Appl. Mat. 35 (1978), 755-786.Google Scholar
11. Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401.Google Scholar
12. Dobrowolski, E., On a question of Lehmer, (to appear).Google Scholar
13. Dufresnoy, J. et Pisot, C., Sur les dérivés successifs d'un Ensemble Fermé d'entiers Algébriques, Bull. Se. Math. Franc. (2) 77 (1973), 129-136.Google Scholar
14. Dufresnoy, J. et Pisot, C., Sur les éléments d'accumulation d'un ensemble fermé d'entiers algébriques, Bull. Se. Math. Franc. (2) 79 (1955), 54-64.Google Scholar
15. Erdélyi, A., “Asymptotic expansions”, Dover, N.Y., 1956.Google Scholar
16. Grandet-Hugot, M., Ensembles fermés d'entiers algébriques, Ann. Se. Éc. Norm. Sup. (3) 82 (1965), 1-35.Google Scholar
17. Kronecker, L., Zwei sàtze ùber gleichungen mit Ganzzahligen coefficienten, Jour, fur Reine und Angew. Math. 53 (1857), 173-175.Google Scholar
18. Lawton, W., Asymptotic properties of roots of polynomials—Preliminary report, Proc. Seventh Iranian National Mathematics Conference, Azarabadegan University, Tabriz, 1976.Google Scholar
19. Lehmer, D. H., Factorization of certain cyclotomic functions, Ann. Math. (2) 34 (1933), 461-479.Google Scholar
20. Lind, D. A., Ergodic automorphisms of the infinite Torus are Bernoulli, Israel J. Math. 17 (1974), 162-168.Google Scholar
21. Mahler, K., On some inequalities for polynomials in several variables, Jour. Lond. Math. Soc. 37 (1962), 341-344.Google Scholar
22. Mendes France, M., Roger Apéry et l'irrationnel, La Recherch. 10 (1979), 170-172.Google Scholar
23. Mignotte, M., Entiers Algébriques dont les conjugés sont proches du cercle unité, Sém. Delange-Pisot-Poitou, 19e année (1977/78), no. 39, (6 p.)Google Scholar
24. Salem, R., A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan, Duke Math. Jour. 11 (1944), 103-108.Google Scholar
25. Salem, R., Power series with integral coefficients, Duke Math. Jour. 12 (1945), 153-172.Google Scholar
26. Schinzel, A., On a problem of Lehmer, Acta Math. Acad. Sci. Hung. (See §7(5)).Google Scholar
27. Siegel, C. L., Algebraic integers whose conjugates lie in the unit circle, Duke Math. Jour. 11 (1944), 597-602.Google Scholar
28. Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London. Math. Soc. 3 (1971), 169-175.Google Scholar
29. Smyth, C. J., A Kronecker-type theorem for complex polynomials in several variables, Canad. Math. Bull., this issue.Google Scholar
30. Stewart, C. L., Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. math. France. 106 (1978), 169-176.Google Scholar
31. Stewart, C. L., On a theorem of Kronecker and a related question of Lehmer, Sém. de théorie des nombres, Bordeaux, 1977/78, no. 7 (11 p.)Google Scholar
32. van der Poorten, A. J., A proof that Euler missed… Apery's proof of the irrationality of £(3), Mathematical Intelligencer. 1 (1978), 195-203.Google Scholar
33. Waldschmidt, M., Sur le produit des conjugués extérieurs au cercle unité d'un entier algébriques, l'enseignement Mathématique (to appear).Google Scholar