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arXiv · 2102.12995

Fast-growing series are transcendental

Robert Dawson·Grant Molnar
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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 $\mathbb{C}[[z]]$.

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Robert Dawson, Grant Molnar. 2021-02-25. Fast-growing series are transcendental. https://arxiv.org/abs/2102.12995

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