MathDBModel-companion conjecture for globally valued fields
Model-companion conjecture for globally valued fields
For , let denote the theory of globally valued fields with Archimedean error . A theory is a model companion of another theory when it is model complete and every model of either theory embeds into a model of the other.
Model-companion conjecture. The theory has a model companion for every Archimedean error .
The existence of such a model companion would provide a model-complete theory governing globally valued fields with each allowed Archimedean error. The paper presents this as an important open question for the development of globally valued fields.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Model companion conjecture for globally valued fields
A globally valued field (GVF) is a field equipped with the global valuation structure considered in the paper. Model companion conjecture. The theory of GVFs has a model companion; equivalently, the ultraproduct of existentially closed GVFs is existentially closed. This is a model-theoretic existence conjecture for GVFs, and the supplied text gives no resolution or partial result beyond stating it.
source: Nuno Hultberg, “New gap principle for semiabelian varieties using globally valued fields”, arXiv:2601.04972 (2026).
Sources & referencesView supporting material
Primary source
Itaï Ben Yaacov, Pablo Destic, Ehud Hrushovski and Michał Szachniewicz, “Globally valued fields: foundations”, arXiv:2409.04570 (2024).
Additional references
2 papers in this index state this conjecture (2018–2024). The statement above is taken from the most recent of them; the others are arXiv:1805.04219.
Progress summary
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