A relative Grace Theorem for complex polynomials

TL;DR

This paper extends classical Grace and Walsh theorems by analyzing how polynomial maps affect the apolarity invariant of complex polynomials, leading to new results involving images of circular domains.

Contribution

It introduces a relative version of Grace's theorem for complex polynomials, considering the pullback of the apolarity invariant under polynomial maps.

Findings

Generalizes Grace and Walsh theorems to polynomial images of circular domains

Provides new bounds and invariance properties for complex polynomials under polynomial maps

Establishes a framework for analyzing polynomial invariants in transformed domains

Abstract

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.