An elementary proof of Takagi's theorem on the differential composition of polynomials

TL;DR

This paper provides a simple, elementary proof of Takagi's 1921 theorem concerning the zeros of composite polynomials formed by differential operators and polynomials.

Contribution

It offers a new, straightforward proof of Takagi's theorem, making the result more accessible and easier to understand.

Findings

Elementary proof of Takagi's theorem presented

Clarifies the structure of zeros in composite polynomials

Simplifies understanding of differential composition of polynomials

Abstract

I give a short and completely elementary proof of Takagi's 1921 theorem on the zeros of a composite polynomial $f(d/dz) \, g(z)$.