ORCID

https://orcid.org/0000-0002-7437-5046

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" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: normal; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">��>0 we prove that for a certain (N+2" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: normal; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">��+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" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: normal; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">1+��/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.

Disciplines

Mathematics | Physical Sciences and Mathematics

DOI

10.1007/s00025-023-02003-4

Full Publication Date

September 2023

Publication Details

A peer-reviewed version of this paper is available in Results in Mathematics, vol. 78, no 229. © Springer Nature.

Publisher

arxiv

Resource Type

preprint

Access Rights

open access

Open Access Route

Green Open Access

License Condition

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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Mathematics Commons

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