A family of polynomials with concyclic zeros

Stolarsky, Kenneth B.

doi:10.1090/S0002-9939-1983-0702287-4

A family of polynomials with concyclic zeros
HTML articles powered by AMS MathViewer

by Kenneth B. Stolarsky PDF

Proc. Amer. Math. Soc. 88 (1983), 622-624 Request permission

Abstract:

Expand $E\left ( z \right ) = {({e^z} - 1)^m}$ by the binomial theorem, and replace every $exp\left ( {{k_z}} \right )$ by its approximation ${\left ( {1 + k{n^{ - 1}}z} \right )^n}$. The resulting polynomial has all of its zeros on a circle of radius $r$ centered at $- r$, where $r = n/m$.

References

Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 0422171
Kenneth L. Cooke, Differential-difference equations, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 155–171. MR 0146481
A. Cohn, Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z. 14 (1922), no. 1, 110–148 (German). MR 1544543, DOI 10.1007/BF01215894
D. G. Dickson, Zeros of exponential sums, Proc. Amer. Math. Soc. 16 (1965), 84–89. MR 171903, DOI 10.1090/S0002-9939-1965-0171903-X
R. E. Langer, On the zeros of exponential sums and integrals, Bull. Amer. Math. Soc. 37 (1931), no. 4, 213–239. MR 1562129, DOI 10.1090/S0002-9904-1931-05133-8
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972

The zeros of a certain family of trinomials

A. J. Van der Poorten, On the number of zeros of functions, Enseign. Math. (2) 23 (1977), no. 1-2, 19–38. MR 476662
A. J. Van der Poorten and R. Tijdeman, On common zeros of exponential polynomials, Enseign. Math. (2) 21 (1975), no. 1, 57–67. MR 379387
David Ruelle, Some remarks on the location of zeroes of the partition function for lattice systems, Comm. Math. Phys. 31 (1973), 265–277. MR 353912
David Ruelle, Extension of the Lee-Yang circle theorem, Phys. Rev. Lett. 26 (1971), 303–304. MR 292441, DOI 10.1103/PhysRevLett.26.303

Geometrisches über die Verteilung der Nullstellen spezieller ganzer funktionen

Kenneth B. Stolarsky, Zero-free regions of exponential sums, Proc. Amer. Math. Soc. 83 (1981), no. 3, 486–488. MR 627675, DOI 10.1090/S0002-9939-1981-0627675-4
K. B. Stolarsky, Zeros of exponential polynomials and “reductionism”, Topics in classical number theory, Vol. I, II (Budapest, 1981) Colloq. Math. Soc. János Bolyai, vol. 34, North-Holland, Amsterdam, 1984, pp. 1539–1552. MR 781194
Marc Voorhoeve, On the oscillation of exponential polynomials, Math. Z. 151 (1976), no. 3, 277–294. MR 430217, DOI 10.1007/BF01214940