On location of zeros of the first derivative

TL;DR

This paper proves that for a complex polynomial, a specific circle centered at the centroid of zeros contains at least half of the zeros of its derivative, providing insight into the zeros' distribution.

Contribution

It establishes a new geometric bound on the location of zeros of the derivative of a polynomial relative to the zeros' centroid.

Findings

Circle centered at zeros' centroid contains at least ⌊(n-1)/2⌋ zeros of the derivative

Provides a geometric criterion for zeros of derivatives of polynomials

Enhances understanding of zeros distribution in complex polynomials

Abstract

Let p(z) be a complex polynomial of degree n. Let C be a circle containing its n-1 zeros, having its center in the centroid of these zeros. We prove that C must contain at least int((n-1):2) zeros of its derivative.