We use level sets to give a counter-example to a conjecture of J. Borcea and B. Shapiro. In particular we show that there are real polynomials P such that there is a chord of the level set Im P′(z)/P(z) = 0 that does not have a non-real zero of P lying on it. In addition, we use level sets to show that there are no bounded components of the set z: Im z > 0 and Im Q(z) > 0 where Q = z− f/f ′ and f is a real entire function.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
J. Borcea and B. Shapiro, Classifying real polynomial pencils, Int. Math. Res. Not. 69 (2004), 3689–3708.
X. Buff, Virtually repelling fixed points, Publ. Mat. 47 no.1 (2003), 195–209.
T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. 125 (1987), 405–431.
S. Edwards, Using level curves to count non-real zeros of f″, in: Advances in Analysis: Proceedings of the Fourth ISAAC Congress, World Scientific Publishing Co., Pte. Ltd., 2005, 531–537.
S. Edwards and S. Hellerstein, Non-real zeros of derivatives of real entire functions and the Pólya-Wiman conjectures, Complex Variables 47 (2002), 25–57.
A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. 155 (2002), 105–129.
C. F. Gauss, Werke 3 (1886), 119–121.
C. F. Gauss, Werke 10 part I (1866), 129–134.
A. Hinkkanen, Reality of zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math. Ser. AI 22 1997) No.1, 21–38.
A. Hinkkanen, Zeros of derivatives of strictly non-real meromorphic functions, Ann. Acad. Sci. Fenn. Math. Ser. AI 22 1997) No.1, 39–74.
A. Hinkkanen, Iteration, level sets, and zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math. Ser. AI 23 1998) No.2, 317–388.
A. Hinkkanen, Zeros of derivatives of meromorphic functions with one pole, Complex Variables 37 (1998), 279–369.
J. Langley, Solution of a Problem of Edwards and Hellerstein, Comp. Methods Funct. Theory 6 no.1 (2006), 243–252.
B. Palka, An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991.
T. Sheil-Small, Complex Polynomials, Cambridge University Press, New York, 2002.
T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Ann. of Math. 129 (1989), 179–193.
M. Tyaglov, On the number of real critical points of logarithmic derivatives and the Hawai’i conjecture, preprint, arXiv:0902.0413.
This material is based upon work supported by the National Science Foundation under Grant No. 0758226. This research was also supported by the grant 07365 from the Campus Research Board of the University of Illinois at Urbana—Champaign.
About this article
Cite this article
Edwards, S., Hinkkanen, A. Level Sets, a Gauss-Fourier Conjecture, and a Counter-Example to a Conjecture of Borcea and Shapiro. Comput. Methods Funct. Theory 11, 1–12 (2011). https://doi.org/10.1007/BF03321786