Residue-Field Domination in Henselian Valued Fields

Definitions and mathematical statement

Let $(K,v)$ be a valued field, with valuation ring $\mathcal O_v$, maximal ideal $\mathfrak m_v$, residue field $k_v=\mathcal O_v/\mathfrak m_v$, and ordered value group $\Gamma_v=v(K^\times)$. Equicharacteristic zero means \[ \operatorname{char}(K)=\operatorname{char}(k_v)=0. \] The valuation is henselian if every simple root in $k_v$ of the reduction of a monic polynomial $f\in\mathcal O_v[X]$ lifts to a root of $f$ in $\mathcal O_v$. The supplied record does not require the valuation to be nontrivial.

Extension problem. Determine hypotheses on henselian valued fields of equicharacteristic zero under which an appropriate notion of residue-field domination extends beyond its original setting, with the intended application of transferring specified tameness properties of the residue field to the valued field. The source asks how far such an extension is possible; it does not supply a conjecture asserting domination for every henselian valued field of equicharacteristic zero. The target classes, language, parameter bases, domination relation, and tameness properties are unspecified, so this is an extension problem rather than a fully quantified conjecture.

Supplied contextual theorem fragment. The record attributes the following assertion to Vicaria as Theorem 4: for $C\subseteq L$, if $C$ is a maximal model of $T$, then $\operatorname{tp}(L/C)$ is dominated by the value group and the residue sorts. Here $\operatorname{tp}(L/C)$ denotes, once a language and ambient structure are fixed, the complete type over $C$ of an enumeration of $L$. The fragment does not specify $T$, the ambient structure, the meaning of maximality, the residue sorts, or the domination relation. Its literal wording says that $C\subseteq L$ are "substructures of $C$", leaving the intended ambient notation unresolved. This incomplete fragment supplies context only and cannot be promoted to a theorem about arbitrary henselian valued fields. Domination by the value group together with residue sorts must also be distinguished from domination by residue-field data alone.

Short English statement

The problem asks how broadly residue-field domination can be extended among henselian valued fields whose field and residue field both have characteristic zero, with the aim of transferring tameness from the residue field. The precise domination notion, hypotheses, and tameness properties are missing. The supplied record labels the problem partially solved; that status is dated curation, not an independently verified conclusion.

Sources

• Supplied local problem-registry record, permanent UnsolvedMath ID $30005292$, source problem code catalogue entry, title: Residue-Field Domination in Henselian Valued Fields. No individual proposer is named; the supplied proposed year is 2022.
• Supplied report citation: Mini-Workshop: Topological and Differential Expansions of o-minimal Structures (2023), Oberwolfach Reports, pp. 3005--3050, DOI 10.4171/owr/2022/52. https://doi.org/10.4171/owr/2022/52. The extracted theorem fragment is attributed to Vicaria. The record labels its reconstructed statement corrected_verified and reports checking it against the official report; these are supplied verification claims, not checks performed for this draft.
• Vicaria, Residue field domination. Supplied literature link: https://arxiv.org/abs/2109.08243.
• Ealy--Haskell--Simon, Residue field domination. Supplied literature link: https://arxiv.org/abs/2212.05188.
• Cubides Kovacsics--Rideau-Kikuchi--Vicaria, Residual domination. Supplied literature link: https://arxiv.org/abs/2410.22712.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports extensions to several classes of henselian equicharacteristic-zero fields, including bounded-Galois-group and maximal-model cases, while describing the broad extension as incomplete. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; its status and cited claims require independent checking.