Strengthening the Gauss-Lucas theorem for polynomials with Zeros in the interior of the convex hull

TL;DR

This paper strengthens the classical Gauss-Lucas theorem by showing that for certain degree four polynomials with zeros forming a concave quadrilateral, the zeros of the derivative are confined to specific regions within the convex hull.

Contribution

It provides a new geometric refinement of the Gauss-Lucas theorem for degree four polynomials with concave zero configurations, extending to non-convex zero arrangements.

Findings

Zeros of the derivative lie in two of the three triangles formed by zeros

The result applies to degree four polynomials with concave quadrilaterals

The theorem can be extended to certain higher-degree polynomials

Abstract

According to the classical Gauss-Lucas theorem all zeros of the derivative of a complex non-constant polynomial p lie in the convex hull of the zeros of p. It is proved that for a polynomial p of degree four with four different zeros forming a concave quadrilateral, the zeros of the derivative lie in two of the three triangles formed by the zeros of p. Thus a strengthening of the classical Gauss-Lucas theorem is established for this case, which can be extended to the case of a polynomial of degree n for which the zeros do not form a convex n-polygon.