Characterization of stably dominated types in infinite imperfection degree

Establish whether, in the theory SCVFp,∞ of separably closed non-trivially valued fields of infinite imperfection degree, the stably dominated types are precisely the types in the natural class Kp,(∞,0)(KtrivΓ,Kdefk), equivalently determine whether the explicitly defined type q of an element with generic residue and p-independent valued-field component is orthogonal to the valued-field sort and is not stably dominated.

Background

For finite imperfection degree, the paper identifies the relevant class of stably dominated types with types whose valued-field extension has relative imperfection degree zero and whose value-group component is trivial. The authors conjecture that an analogous characterization holds in infinite imperfection degree, but this requires understanding a type involving a p-independent valued-field component.

Specifically, the type q is defined over a model K of SCVF_p,∞ by taking an element x in the valuation ring whose residue is generic over the residue field k(K), while x is p-independent over the valued field VF(K). The conjecture asserts both that q is orthogonal to the valued-field sort (equivalently, generically stable) and that q is not stably dominated. The paper notes that this conjecture would imply that SCVF_p,∞ is not metastable over the value group.

References

We conjecture that the stably dominated types in $SCVF_{p,\infty}$ are precisely the types in the class $K_{p,(e,0)}(K{triv}{},K{def}{k})$. This is easily seen to be equivalent to Conjecture~\ref{Conj:stdom} below. Contrarily to the case of finite imperfection degree $e$, where the imaginaries in $SCVF_{p,e}$ are classified by the geometric sorts (see ), no classification of imaginaries in $SCVF_{p,\infty}$ is known, and so an approach by inspection in the (geometric) sorts is not possible.

— On separable pairs of fields and beautiful pairs of separably closed valued fields  (2610.18753 - He et al., 13 Oct 2026) in Section 12, paragraph immediately preceding Conjecture \ref{Conj:stdom}; formal statement in Conjecture \ref{Conj:stdom}

Let $K\modelsSCVF_{p,\infty}$, and let $q(x)$ be the definable type of an element $x\inO()$ such that $res(x)$ is generic over $k(K)$ and $x$ is $p$-independent in $VF()$ over $VF(K)$. Then $q$ is orthogonal to $VF$ (equivalently, generically stable). We conjecture that $q$ is not stably dominated.

— On separable pairs of fields and beautiful pairs of separably closed valued fields  (2610.18753 - He et al., 13 Oct 2026) in Conjecture \ref{Conj:stdom}, Section 12