Roots of Polynomials and Umbilics of Surfaces

TL;DR

This paper establishes a bound on the number of roots inside the unit circle for certain polynomials by relating it to the index of umbilic points on convex surfaces, using geometric and topological methods.

Contribution

It introduces a novel connection between polynomial roots and umbilic points on surfaces, providing a new bound for polynomials with self-inversive second derivatives.

Findings

Number of roots inside the unit circle ≤ 1 + N/2 for specific polynomial families.

Constructs convex surfaces from polynomials to relate roots to umbilic indices.

Bound on roots derived from Hamburger's bound on umbilic point indices.

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.