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Proceedings of the American Mathematical Society

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Sums of powers of conjugates of algebraic numbers

Author: P. E. Blanksby
Journal: Proc. Amer. Math. Soc. 49 (1975), 28-32
MSC: Primary 10F30; Secondary 12A15
MathSciNet review: 0382180
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Abstract: This note is primarily concerned with algebraic numbers which have at least one conjugate with modulus exceeding one. Theorems which give lower bounds for the maximum of the moduli of the conjugates of such algebraic numbers are connected with equivalent theorems giving lower bounds for the sums of powers of the conjugates. The results relate to some earlier work of S. Chowla, and the method used depends on one of P. Turán's main theorems on lower bounds for sums of powers of complex numbers.

References [Enhancements On Off] (What's this?)

  • [1] P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369. MR 0296021
  • [2] S. Chowla, On polynomials all of whose roots lie on the unit circle, J. Reine Angew. Math. 222 (1966), 69–70. MR 0188165
  • [3] L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.
  • [4] Paul Turán, Eine neue Methode in der Analysis und deren Anwendungen, Akadémiai Kiadó, Budapest, 1953 (German). MR 0060548
  • [5] A. J. Van der Poorten, Generalisations of Turán’s main theorems on lower bounds for sums of powers, Bull. Austral. Math. Soc. 2 (1970), 15–37. MR 0265307

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Keywords: Algebraic numbers, conjugates
Article copyright: © Copyright 1975 American Mathematical Society