Polynomials in one variable and ranks of certain tangent maps

TL;DR

This paper investigates a specific map from monic degree n complex polynomials without multiple roots to the collection of derivative values at roots, addressing a question posed by Ju.S. Ilyashenko.

Contribution

It provides a detailed analysis of the tangent map associated with polynomials and answers an open question in the field.

Findings

Characterization of the tangent map for monic polynomials

Solution to Ilyashenko's question about polynomial derivatives

Insights into the structure of polynomial root derivatives

Abstract

We study a map that sends a monic degree n complex polynomial f(x) without multiple roots to the collection of n values of its derivative at the roots of f(x). We give an answer to a question posed by Ju.S. Ilyashenko.