Fast-growing series are transcendental

TL;DR

The paper proves that rapidly growing power series coefficients lead to transcendental series over certain rings, providing an alternative proof of a known result using relationships between polynomial coefficients.

Contribution

It offers a new proof of the transcendence of series with rapidly growing coefficients, connecting coefficient growth to algebraic independence over rings.

Findings

Rapid coefficient growth implies transcendence.

Established a relationship between coefficients of $A(X)$ and $A'(X)$.

Provided an alternative proof of the Newton-Puiseux Theorem.

Abstract

Let $R$ be a subring of $\mathbb{C}[[z]]$, and let $X \in \mathbb{C}[[z]]$. The Newton-Puiseux Theorem implies that if the coefficients of $X$ grow sufficiently rapidly relative to the coefficients of the series in $R$, then $X$ is transcendental over $R$. We prove an alternative proof of this result by establishing a relationship between the coefficients of $A(X)$ and $A^\prime(X)$, where $A(T)$ is a polynomial over $\mathbb{C}[[z]]$.