On a paper by Y.G.Zarhin

TL;DR

This paper revisits Zarhin's result on the co-rank of a matrix derived from a polynomial with distinct roots, providing a direct proof, exploring the case of multiple roots, and proposing a conjecture on its rank.

Contribution

It offers a direct proof of Zarhin's theorem and investigates the properties of the matrix for polynomials with multiple roots, including a new conjecture.

Findings

Matrix has co-rank one for polynomials with distinct roots

Direct proof of Zarhin's result is provided

Conjecture about the rank for polynomials with multiple roots

Abstract

Zarhin showed that a matrix constructed from a polynomial with distinct roots has co-rank one. Some striking properties of this matrix are used to give a direct proof of his result. An account is given of calculations carried out to try to understand the analogous matrix for a polynomial with multiple roots and a conjecture about its rank is stated.