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On measures of polynomials in several variables

Published online by Cambridge University Press:  17 April 2009

C.J. Smyth
C.J. Smyth
Affiliation:
Department of Mathematics, James Cook University, Townsville, Queensland 4811, Australia.
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Abstract

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The measure of a polynomial is defined as the exponential of a certain intractable-looking integral. However, it is shown how the measures of certain polynomials can be evaluated explicitly: when all their irreducible factors are linear, and belong to one of two special classes. Asymptotic values for the measures of two sequences of polynomials in large numbers of variables are also found. The proof of this result uses a quantitative form of the central limit theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 1 , February 1981 , pp. 49 - 63
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Bhattacharya, R.N. and Rao, R. Ranga, Normal approximations and asymptotic expansions (John Wiley & Sons, New York, London, Sydney, 1976).Google Scholar
[2]Boyd, David W., “Kronecker's theorem and Lehmer's problem for polynomials in several variables”, J. Number Theory (to appear).Google Scholar
[3]Boyd, David W., “Speculations concerning the range of Mahler's measure”, Canad. Math. Bull. (to appear).Google Scholar
[4]Lawton, Wayne M., “A generalization of a theorem of Kronecker”, J. Sci. Fac. Chiang Mai Univ. 4 (1977), 15–23.Google Scholar
[5]Lewin, L., Dilogarithms and assooiated functions (Macdonald, London, 1958).Google Scholar
[6]Mahler, K., “On some inequalities for polynomials in several variables”, J. London Math. Soc. 37 (1962), 341–344.Google Scholar
[7]Smyth, C.J., “A Kronecker-type theorem for complex polynomials in several variables”, Canad. Math. Bull. (to appear).Google Scholar