Roots of Polynomials and Umbilics of Surfaces

Authors

Brendan Guilfoyle, School of STEM, Munster Technological University, Tralee, County Kerry, IrelandFollow
Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK

ORCID

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

Document Type

Preprint

Disciplines

Mathematics | Physical Sciences and Mathematics

Publication Details

Results in Mathematics, vol. 78, no 229. © Springer Nature.

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;">𝑁>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;">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" 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+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.

Recommended Citation

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