Geometrically trivial strongly minimal sets that fail $\aleph_0$-categoricity: classification beyond arithmetic groups

An old question of Hrushovski (ICM 1998) asked whether every geometrically trivial strongly minimal set definable in a differentially closed field of characteristic zero is $\aleph_0$-categorical. Freitag and Scanlon refuted it with the third-order differential equation of the modular $j$-function, in which elliptic-curve isogenies force algebraic relations of unbounded degree among generic solutions; work of Casale, Freitag and Nagloo ties every known counterexample to arithmetic Fuchsian groups, and their Question 2.15 first posed the search for an example outside that family. As restated by Freitag, León Sánchez, Li and Nagloo (arXiv:2603.12387, 2026), the open problem is to find a geometrically trivial strongly minimal set, orthogonal to the constants, that fails $\aleph_0$-categoricity for reasons unrelated to arithmetic group phenomena; the same paper records that a structural 'revision of Hrushovski's question' accounting for the known failures 'would be of great interest' and that 'to date, one has not been made'.

Background

A differential field $(K,\delta)$ of characteristic zero has a field of constants $\mathcal{C}=\{x\in K:\delta x=0\}$. The theory $\mathrm{DCF}_0$ of differentially closed fields, the model companion of the theory of characteristic-zero differential fields, is the standard model-theoretic setting for algebraic differential equations: every consistent finite system of differential polynomial equations has a solution in a model, and solution sets of such systems are exactly the definable sets. A definable set $X$ is strongly minimal if every definable subset of $X$ is finite or cofinite; its generic solutions carry a dimension (Morley rank one), and algebraic closure induces on $X$ a pregeometry $a\mapsto\operatorname{acl}(a)\cap X$ which, by a theorem of Hrushovski, is trivial, locally modular, or non-locally-modular.

Three notions from geometric stability theory are needed. $X$ is geometrically trivial if whenever $a,b,c\in X$ and $c\in\operatorname{acl}(a,b)$ then $c\in\operatorname{acl}(a)$ or $c\in\operatorname{acl}(b)$: algebraic dependencies among solutions never genuinely involve two solutions at once. $X$ is orthogonal to the constants ($X\perp\mathcal{C}$) if no generic solution of $X$ becomes definable over constants in any extension; sets internal to $\mathcal{C}$ are the opposite, degenerate extreme. A strongly minimal set is $\aleph_0$-categorical if, over a fixed base, there are only finitely many complete $n$-types of tuples from $X$ for each $n$; for a geometrically trivial strongly minimal set this is equivalent to a uniform finite bound on $|\mathcal{C}\langle x\rangle^{alg}\cap X|$ as $x$ ranges over non-constant solutions (the formulation used by Duan and Nagloo). Hence failure of $\aleph_0$-categoricity for a trivial set means precisely that its generic solutions satisfy algebraic relations of unbounded degree.

For order-one equations $f(y,\delta y)=0$ over a differential field $k$, Kumbhakar, Roy and Srinivasan (2024) classify equations into algebraic type (all solutions algebraic over $k$), Riccati type (a generic solution algebraic over a solution of a Riccati equation $x'=a_2x^2+a_1x+a_0$ with coefficients in $k$), Weierstrass type (a generic solution algebraic over a solution of $(\delta x)^2=\alpha^2(4x^3-g_2x-g_3)$), and general type (the remaining case), and they conjectured that an equation is not of general type if and only if it has at most three algebraically independent solutions in any differential field extension of $k$ with the same constants (at most one, in the autonomous case, when the base field is its own constant field). Freitag, León Sánchez, Li and Nagloo (2026) prove this conjecture and prove, for every order $n$, the higher-order conjecture of Kumbhakar and Srinivasan (2025) — the set $\mathfrak{S}_{k,f,y}$ of mutually $k$-isomorphic independent generic solutions in constant-preserving extensions is finite if and only if it has at most $n+2$ elements (at most one over a constant field) — with optimality of the bounds established via the inverse problem in differential Galois theory (realizing a prescribed linear algebraic group as a differential Galois group), using an appendix by Michael F. Singer. The proofs rest on binding groups (automorphism groups of a generic type fixing the base and the constants, identified with the Galois groups of Kolchin's strongly normal extensions — the differential analogue of Galois extensions), their definable group actions, and the Borovik–Cherlin bound (an upper bound on the generic transitivity degree of a definable group action in terms of the group's Morley rank), from work of Freitag–Jaoui–Moosa, Freitag–Moosa, and Freitag–Jimenez–Moosa.

The present problem concerns the trivial end of this classification. The Hrushovski–Sokolović trichotomy (unpublished; recorded as Theorem 5.2 of Casale–Freitag–Nagloo and in Section 4 of the 2026 paper) says that every strongly minimal set in $\mathrm{DCF}_0$ is exactly one of the following: nonorthogonal to $\mathcal{C}$; nonorthogonal to a Manin kernel — the differential-algebraic kernel of a simple abelian variety not defined over the constants, of which the sixth Painlevé equation (a classical nonlinear differential equation governing the universal elliptic curve) is the universal instance, where multiplication-by-$n$ maps supply algebraic relations of unbounded degree among generic solutions; or geometrically trivial. In particular, a geometrically trivial strongly minimal set is automatically orthogonal to $\mathcal{C}$. This left the question Hrushovski posed at the 1998 International Congress of Mathematicians: is every geometrically trivial strongly minimal set $\aleph_0$-categorical? Freitag and Scanlon (2017) refuted it: the third-order differential equation satisfied by the modular $j$-function (the invariant parametrizing isomorphism classes of elliptic curves, whose Schwarzian derivative $S(f)=f'''/f'-\frac{3}{2}(f''/f')^{2}$ obeys a third-order algebraic differential equation) defines a strongly minimal, geometrically trivial set that is not $\aleph_0$-categorical, because isogenies — surjective algebraic group homomorphisms with finite kernel — between elliptic curves produce algebraic relations between $j(a)$ and $j(a')$ of unbounded degree. Casale, Freitag and Nagloo (2020) proved Ax–Lindemann–Weierstrass with derivatives (a functional-transcendence statement describing the algebraic relations between a uniformizing function and its derivatives) for uniformizing functions of genus-zero Fuchsian groups — discrete subgroups of $\mathrm{PSL}_2(\mathbb{R})^+$ acting on the upper half-plane with genus-zero quotient, the arithmetic ones being those commensurable with the unit group of an order in a quaternion algebra over a totally real field, $\mathrm{PSL}_2(\mathbb{Z})$ being the basic example. Within this Fuchsian family they showed that the equation of the uniformizer $j_\Gamma$ is $\aleph_0$-categorical if and only if $\Gamma$ is not arithmetic, and they posed, as the next major challenge in the classification of geometrically trivial strongly minimal sets (their Question 2.15), whether there are non-$\aleph_0$-categorical strongly minimal sets that do not arise from arithmetic Fuchsian groups — where, per their footnote, 'arising from' means nonorthogonal to the Schwarzian equation of an arithmetic Fuchsian uniformizer or one of its other fibers. As both the 2026 paper and Duan–Nagloo record, every presently known instance of a geometrically trivial strongly minimal equation failing $\aleph_0$-categoricity is so related. The 2026 paper closes its optimality discussion by stating that a revision of Hrushovski's question 'would be of great interest' and that 'to date, one has not been made', and that it is 'an interesting open problem' to find such a strongly minimal set unrelated to arithmetic groups.

The order-by-order picture as of 2026: for order one, geometric triviality implies $\aleph_0$-categoricity (unpublished work of Hrushovski from 1995, as recounted by Duan and Nagloo). For order two, Duan and Nagloo conjecture that every strongly minimal set definable over the constants is $\omega$-categorical (their notation; the same notion as $\aleph_0$-categorical) and prove it for equations of Brestovski form — order-two equations of the shape $F'=\sum_{i=1}^{n}a_iG_i'/G_i$ with $F$ a differential polynomial and the $G_i$ differential rational functions over the constants, the $a_i$ linearly independent over $\mathbb{Q}$. They also prove that the classical Lotka–Volterra system $x'=axy+bx$, $y'=cxy+dy$ (a planar autonomous system arising in population dynamics) is strongly minimal and geometrically trivial whenever $d$ and $b$ are linearly independent over $\mathbb{Q}$ — a new, non-arithmetic source of trivial strongly minimal sets, though one that is $\omega$-categorical — and Duan, Eagles and Jimenez extend the analysis to multiple such systems. All known counterexamples are of order three and arithmetic-Fuchsian in origin. What is missing is any account, structural or constructive, of the non-$\aleph_0$-categorical trivial sets beyond the arithmetic-group family.

Problem Statement

The problem, as posed in Section 4 of Freitag–León Sánchez–Li–Nagloo (arXiv:2603.12387) — and, earlier, as Question 2.15 of Casale–Freitag–Nagloo in the context of the classification of geometrically trivial strongly minimal sets — is to determine whether arithmetic-group phenomena are the only source of failure of Hrushovski's principle that geometrically trivial strongly minimal sets in differentially closed fields of characteristic zero are $\aleph_0$-categorical. Work in a sufficiently saturated model of $\mathrm{DCF}_0$ with one derivation and field of constants $\mathcal{C}$.

(1) New example. Construct a definable strongly minimal set $X$ such that: (i) $X$ is geometrically trivial; (ii) $X$ is orthogonal to $\mathcal{C}$ (automatic given (i), by the Hrushovski–Sokolović trichotomy; it is retained because the source's discussion is framed for sets orthogonal to the constants); (iii) $X$ is not $\aleph_0$-categorical — equivalently, there is no uniform finite bound on $|\mathcal{C}\langle x\rangle^{alg}\cap X|$ over non-constant solutions $x\in X$, i.e., the generic solutions of $X$ satisfy algebraic relations of unbounded degree; and (iv) $X$ does not arise from arithmetic Fuchsian groups, in the sense fixed by the footnote to Question 2.15 of Casale–Freitag–Nagloo: $X$ is not nonorthogonal to the Schwarzian equation of an arithmetic Fuchsian uniformizer or to one of its other fibers. Condition (iv) is non-vacuous and separates the requested example from the known ones: the known counterexamples — the $j$-function equation of Freitag–Scanlon and, more generally, the equations of arithmetic Fuchsian uniformizers — do arise from arithmetic Fuchsian groups in this sense.

(2) Classification. Prove that no set satisfying (i)–(iv) exists, by establishing a structural theorem for all geometrically trivial strongly minimal sets in $\mathrm{DCF}_0$: failure of $\aleph_0$-categoricity implies arising from arithmetic Fuchsian groups in the sense of condition (iv) (for example, definable nonorthogonality to the Schwarzian equation of an arithmetic Fuchsian uniformizer or one of its other fibers). Since the source question asks whether such sets exist, this branch must match that question in full generality; the source paper records that a 'revision of the question of Hrushovski' along such structural lines 'would be of great interest', that 'to date, one has not been made', and that the work of Casale–Freitag–Nagloo 'provides a suggestion'.

Either branch, if correct, closes the problem. Results confined to special subclasses — for instance, a resolution of the Duan–Nagloo order-two $\omega$-categoricity conjecture — address particular classes of strongly minimal sets, whereas the question concerns all geometrically trivial strongly minimal sets.

References

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