Dimension-Independent Nonminimality of Difference Equations

Definitions and mathematical statement

A difference field is a field equipped with a distinguished automorphism $\sigma$. The theory $\mathrm{ACFA}$ is the model companion of the theory of difference fields. Let $K\subseteq\mathcal U\models\mathrm{ACFA}$ be a difference subfield, with the automorphism on $K$ given by restriction. For a variety with a specified field of definition, its $\sigma$-twist $V^\sigma$ is obtained by applying $\sigma$ to the coefficients of its defining equations. The supplied formulation considers an algebraic variety $V$ and a rational map $\phi:V\dashrightarrow V^\sigma$ over $K$. A solution is a point $a\in V(\mathcal U)$ in the domain of definition of $\phi$ satisfying \[ \sigma(a)=\phi(a), \] where $\sigma$ acts coordinatewise. Write $p=\operatorname{tp}(a/K)$ for the complete type of $a$ over $K$.

Intended conjecture, with unresolved source qualifications. For every such choice of data and every generic solution $a$ whose type $p$ is nonminimal, is \[ \operatorname{nmdeg}(p)\leq 3 \] true, with the same constant $3$ for every dimension $\dim V$? Here $\operatorname{nmdeg}$ denotes the degree of nonminimality named in the supplied record.

The supplied material does not define this invariant or specify the conventions for minimality and genericity in $\mathrm{ACFA}$. Consequently, the displayed assertion records the intended target but does not supply a complete intrinsic definition of its predicate or invariant. The field of definition of $V$, any irreducibility or dominance hypotheses, and any additional hypotheses on $K$, the type, or the characteristic must also be checked against the report. None are silently imposed here. The retained report fragment proposes replacing an earlier expression $\dim(V)+3$ by $3$, but omits the preceding assertion and its hypotheses; that earlier bound is therefore not asserted here as a theorem.

Short English statement

The supplied conjecture asks whether every nonminimal generic solution type of an algebraic difference equation has degree of nonminimality at most $3$, independently of the dimension of the underlying variety. The definitions of the invariant and the precise genericity and hypothesis conventions require source review before this becomes a fully specified mathematical statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006583$, titled Dimension-Independent Nonminimality of Difference Equations; exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2026$, without identifying a proposer.
• Supplied report citation: Model Theory and Arithmetic (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/3, pp. 157--216. Source URL: https://doi.org/10.4171/owr/2026/3. The supplied literature_sources entry is labelled "Oberwolfach Report 3/2026" and links to https://doi.org/10.4171/owr/2026/3.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the report presents the dimension-independent bound as conjectural and gives no proof. This is supplied dated curation, not an independently verified literature assessment.
• The registry assigns the statement status corrected_verified and reports that the official report was checked directly and that OCR, notation, and source-dependent wording were repaired. These are supplied provenance claims; this draft does not independently verify them.
• The retained original passage mentions a conjecture attributed to Poizat concerning minimality of almost every nonlinear equation, then proposes replacing $\dim(V)+3$ by $3$. Its reference "[2]" is unresolved in the supplied material. The mention of Poizat does not identify the proposer of the present bound.