Abstract
For certain polynomials we relate the number of roots inside the unit circle with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In particular, for \(N>0\) we prove that for a certain (\(N+2\))-real dimensional family of complex polynomials of degree N, the number of roots inside the unit circle is less than or equal to \(1+N/2\). This bound is established as follows. From the polynomial we construct a convex real analytic surface containing an isolated umbilic point, such that the index of the umbilic point is determined by the number of roots of the polynomial that lie inside the unit circle. The bound on the number of roots then follows from Hamburger’s bound on the index of an isolated umbilic point on a convex real analytic surface. The class of polynomials that arise are those with self-inversive second derivative. Thus the number of roots inside the unit circle is proven to be bounded for a polynomial with self-inversive second derivative.
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Acknowledgements
The authors would like to thank Gerhard Schmeisser for helpful discussions during the original evolution of this work and to the anonymous Referee for pointing out the relationship with self-inversive polynomials.
Funding
This research was supported by the Research in Pairs program of the Mathematisches Forschungsinstitut Oberwolfach. The authors have no further relevant financial or non-financial interests to disclose. No data was collected in the course of this research.
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Guilfoyle, B., Klingenberg, W. Roots of Polynomials and Umbilics of Surfaces. Results Math 78, 229 (2023). https://doi.org/10.1007/s00025-023-02003-4
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DOI: https://doi.org/10.1007/s00025-023-02003-4