On the problem of Kakeya

TL;DR

This paper extends classical results on the Kakeya problem for complex polynomials by establishing conditions under which a disc containing certain zeros of a polynomial's derivatives can be determined, generalizing the Grace-Heawood theorem.

Contribution

It introduces new bounds for locating zeros of derivatives of complex polynomials based on known zero-containing discs, extending existing theorems.

Findings

Established a disc containing a zero of the (k-1)-th derivative given a disc with k zeros of p(z)

Generalized the Grace-Heawood theorem for higher derivatives

Provided a new approach for zero localization in complex polynomial derivatives

Abstract

We discuss a form of a well-known problem of Kakeya for complex polynomials. Let p(z) be a complex polynomial. This problem requires to find disc that contains n zeros of some derivative of p(z), provided that location of several zeros of p(z) is known. We find a disc that contains a zero of (k-1)-th derivative, if we know disc that contains k zeros of p(z). This result can be regarded as an extension of the Grace-Heawood theorem where k=2.