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PreprintCC BY-NC-ND 4.0HAL

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

  • Stéphane Fischler
  • Tanguy Rivoal
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Abstract

Let E be the ring of values taken at algebraic points by E-functions. The exponential function and the Bessel function J0 are classical examples of E-functions. We prove that E is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of E explicity, including e+1, J0(1), and 1F1(a;b;1) for rational parameters a and b such that a∈ (0,1) and b−a∈ (0,1).

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Cite

BibTeXRIS
BibTeX
@misc{fischler2026ring,
  title = {{The ring of values of $E$-functions is not a field}},
  author = {Stéphane Fischler and Tanguy Rivoal},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2610.16603},
  eprint = {2610.16603},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05763493v2},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The ring of values of E-functions is not a field
AU  - Stéphane Fischler
AU  - Tanguy Rivoal
PY  - 2026
DA  - 2026/09/24
PB  - HAL
DO  - 10.48550/arxiv.2610.16603
UR  - https://hal.science/hal-05763493v2
AB  - Let 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 E is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of E explicity, including e+1, J_0(1), and _1F_1(a;b;1) for rational parameters a and b such that a∈ (0,1) and b−a∈ (0,1).
ER  -