The ring of values of $E$-functions is not a field

Stéphane Fischler; Tanguy Rivoal

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arXiv · 2610.16603

The ring of values of $E$-functions is not a field

Stéphane Fischler·Tanguy Rivoal

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Abstract

Let $\mathbf{E}$ be the ring of values taken at algebraic points by $E$-functions. The exponential function and the Bessel function $J_0$ are classical examples of $E$-functions. We prove that $\mathbf{E}$ is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of $\mathbf{E}$ explicity, including $e+1$, $J_0(1)$, and ${}_1F_1(a;b;1)$ for rational parameters $a$ and $b$ such that $0

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BibTeX RIS

Stéphane Fischler, Tanguy Rivoal. 2026-10-12. The ring of values of $E$-functions is not a field. https://arxiv.org/abs/2610.16603

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