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PreprintCC BY 4.0arXiv

Diophantine definability from addition and divisibility over rings of S-integers

  • Carlos Martínez-Ranero
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Abstract

Let K be a number field, let S be a finite set of places of K containing the archimedean places, and let 𝒪K,S denote the ring of S-integers of K. We prove that multiplication is positive-existentially definable in (𝒪K,S,0,1,+,|) if and only if the unit group 𝒪K,S× is infinite. This extends a 1977 theorem of Lipshitz for rings of integers to rings of S-integers. One of the key ingredients is a uniform bound for unit equations due to Evertse, Schlickewei and Schmidt, which we use to recover multiplication on the unit group.

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Cite

BibTeXRIS
BibTeX
@misc{martinezranero2026diophantine,
  title = {{Diophantine definability from addition and divisibility over rings of $S$-integers}},
  author = {Carlos Martínez-Ranero},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05687},
  eprint = {2611.05687},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05687},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Diophantine definability from addition and divisibility over rings of S-integers
AU  - Carlos Martínez-Ranero
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05687
UR  - https://arxiv.org/abs/2611.05687
AB  - Let K be a number field, let S be a finite set of places of K containing the archimedean places, and let 𝒪_K,S denote the ring of S-integers of K. We prove that multiplication is positive-existentially definable in (𝒪_K,S,0,1,+,|) if and only if the unit group 𝒪_K,S^× is infinite. This extends a 1977 theorem of Lipshitz for rings of integers to rings of S-integers. One of the key ingredients is a uniform bound for unit equations due to Evertse, Schlickewei and Schmidt, which we use to recover multiplication on the unit group.
ER  -