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arXiv · 2611.05687

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 $\mathcal{O}_{K,S}$ denote the ring of $S$-integers of $K$. We prove that multiplication is positive-existentially definable in $(\mathcal{O}_{K,S},0,1,+,\mid)$ if and only if the unit group $\mathcal{O}_{K,S}^\times$ 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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Carlos Martínez-Ranero. 2026-11-03. Diophantine definability from addition and divisibility over rings of $S$-integers. https://arxiv.org/abs/2611.05687

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