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Lehmer's problem for compact Abelian groups
Author(s):
Douglas
Lind
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1411-1416.
MSC (2000):
Primary 43A40, 22D40;
Secondary 37B40, 11G50
Posted:
October 18, 2004
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Abstract:
We formulate Lehmer's Problem concerning the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components, then its Lehmer constant vanishes.
References:
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- Tom M. Apostol, Resultants of cyclotomic polynomials, Proc. Amer. Math. Soc. 24 (1970), 457-462. MR 0251010 (40:4241)
- 2.
- David W. Boyd, Speculations concerning the range of Mahler's measure, Canad. Math. Bull. 24 (1981), no. 4, 453-469. MR 0644535 (83h:12002)
- 3.
- David W. Boyd, Reciprocal polynomials having small measure II, Math. Comp. 53 (1989), 353-357, S1-S5. MR 0968149 (89m:30013)
- 4.
- Graham Everest and Thomas Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, New York, 1999. MR 1700272 (2000e:11087)
- 5.
- Wayne M. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983), 356-362. MR 0707608 (84i:10056)
- 6.
- D. H. Lehmer, Factorization of certain cyclotomic functions, Annals of Math. 34 (1933), 461-479. MR 1503118
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Additional Information:
Douglas
Lind
Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195--4350
Email:
lind@math.washington.edu
DOI:
10.1090/S0002-9939-04-07753-6
PII:
S 0002-9939(04)07753-6
Keywords:
Lehmer's Problem,
Mahler measure,
compact abelian group
Received by editor(s):
September 12, 2003
Received by editor(s) in revised form:
January 7, 2004
Posted:
October 18, 2004
Communicated by:
Wen-Ching Winnie Li
Copyright of article:
Copyright
2004,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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