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A family of polynomials with concyclic zeros. II


Authors: Ronald J. Evans and Kenneth B. Stolarsky
Journal: Proc. Amer. Math. Soc. 92 (1984), 393-396
MSC: Primary 30C15; Secondary 33A10
DOI: https://doi.org/10.1090/S0002-9939-1984-0759660-9
MathSciNet review: 759660
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\lambda _1}, \ldots ,{\lambda _J}$ be nonzero real numbers. Expand

$\displaystyle E(z) = \prod {( - 1 + \exp {\lambda _j}z)} ,$

rewrite products of exponentials as single exponentials, and replace every $ \exp (az)$ by its approximation $ {(1 + a{n^{ - 1}}z)^n}$, where $ n \geqslant J$. The resulting polynomial has all zeros on the (possibly infinite) circle of radius $ \left\vert r \right\vert$ centered at $ - r$, where $ r = n/\sum {\lambda _j}$.

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

  • [C] A. Cohn, Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z. 14 (1922), 110-148. MR 1544543
  • [DeB] N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197-226. MR 0037351 (12:250a)
  • [I] L. Ilieff, Über die Nullstellen einiger Klassen von Polynomen, Tôhoku Math. J. 45 (1939), 259-264.
  • [L-S] E. H. Lieb and A. D. Sokal, A general Lee-Yang theorem for one-component and multicomponent ferromagnets, Comm. Math. Phys. 80 (1981), 153-179. MR 623156 (83c:82008)
  • [O] N. Obrechkoff, Sur les racines des équations algébriques, Tôhoku Math. J. 38 (1933), 93-100.
  • [S1] K. B. Stolarsky, A family of polynomials with concyclic zeros, Proc. Amer. Math. Soc. 88 (1983), 622-624. MR 702287 (84m:30010)
  • [S2] -, Zeros of exponential polynomials and "reductionism", Topics in Classical Number Theory, Colloq. Math. Soc. János Bolyai, vol. 34, Elsevier, North-Holland (to appear). MR 781194 (86e:11083)
  • [S3] -, A family of polynomials with concyclic zeros. III (in preparation).
  • [T] E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford, 1951. MR 0046485 (13:741c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0759660-9
Keywords: Concyclic zeros, exponential, exponential polynomial, linear fractional transformations, zeros of polynomials
Article copyright: © Copyright 1984 American Mathematical Society

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