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 1Other wordings
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).
References
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.
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