Sums of powers of conjugates of algebraic numbers
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- by P. E. Blanksby PDF
- Proc. Amer. Math. Soc. 49 (1975), 28-32 Request permission
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
- P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369. MR 296021, DOI 10.4064/aa-18-1-355-369
- S. Chowla, On polynomials all of whose roots lie on the unit circle, J. Reine Angew. Math. 222 (1966), 69–70. MR 188165, DOI 10.1515/crll.1966.222.69 L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.
- Paul Turán, Eine neue Methode in der Analysis und deren Anwendungen, Akadémiai Kiadó, Budapest, 1953 (German). MR 0060548
- 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 265307, DOI 10.1017/S0004972700041575
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 49 (1975), 28-32
- MSC: Primary 10F30; Secondary 12A15
- DOI: https://doi.org/10.1090/S0002-9939-1975-0382180-5
- MathSciNet review: 0382180