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On measures of polynomials in several variables

  • C.J. Smyth (a1)
Abstract

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.

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References
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[1]Bhattacharya R.N. and Rao R. Ranga, Normal approximations and asymptotic expansions (John Wiley & Sons, New York, London, Sydney, 1976).
[2]Boyd David W., “Kronecker's theorem and Lehmer's problem for polynomials in several variables”, J. Number Theory (to appear).
[3]Boyd David W., “Speculations concerning the range of Mahler's measure”, Canad. Math. Bull. (to appear).
[4]Lawton Wayne M., “A generalization of a theorem of Kronecker”, J. Sci. Fac. Chiang Mai Univ. 4 (1977), 1523.
[5]Lewin L., Dilogarithms and assooiated functions (Macdonald, London, 1958).
[6]Mahler K., “On some inequalities for polynomials in several variables”, J. London Math. Soc. 37 (1962), 341344.
[7]Smyth C.J., “A Kronecker-type theorem for complex polynomials in several variables”, Canad. Math. Bull. (to appear).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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