Statement $(C_2)$: binary algebraic independence of the solutions of a general planar algebraic vector field

For a general (coefficientwise generic) planar algebraic vector field of degree $d \ge 3$, the set of solutions in a differentially closed field is known to be strongly minimal and disintegrated, and by follow-up work of Jaoui (2023–2025) the same holds for generic polynomial systems in $\mathbb{A}^n$ for every $n, d \ge 2$ and for ample families on other base varieties. The surviving open core of Jaoui 2021 is the terminal classification step, statement $(C_2)$ of Jaoui's conjecture: prove or refute that any two distinct non-stationary analytic solutions of a general planar algebraic vector field of sufficiently large degree are algebraically independent over $\mathbb{C}$. A positive answer would show that the structure induced on the solution set is exactly a strictly disintegrated (structureless) infinite set expanded by $d^2$ constants naming the singularities, completing the classification of all algebraic relations among the solutions of general second-order algebraic differential equations.

Background

A complex planar algebraic vector field of degree $d$ is a pair of polynomials $f, g \in \mathbb{C}[x,y]$ of degree at most $d$, viewed as the autonomous differential system $\frac{dx}{dt} = f(x,y)$, $\frac{dy}{dt} = g(x,y)$. The vector field is called general (or generic) if the finitely many coefficients of $f$ and $g$ are algebraically independent over $\mathbb{Q}$; equivalently, the desired property is required to hold for all parameter values outside a countable union of proper Zariski-closed subsets of the coefficient space. The model-theoretic home for studying its solutions is $\mathrm{DCF}_0$, the theory of differentially closed fields of characteristic zero (existentially closed differential fields, in which every differential equation with a solution somewhere has a solution). A definable set is strongly minimal if every definable subset is finite or cofinite; for the solution set of the system this means every solution in any differential field extension has transcendence degree $0$ or $2$ over that field. A strongly minimal set is called disintegrated, or geometrically trivial, when every algebraic dependence among finitely many realizations is already generated by dependences between pairs of them; informally, the combinatorial geometry attached to the set is trivial.

Since the 1970s, culminating in questions of Bruno Poizat from the early 1980s (as recounted in Jaoui 2021 and in DeVilbiss–Freitag 2023), the expectation has been that a sufficiently general nonlinear differential equation of order at least two should define a strongly minimal, disintegrated set of solutions — the generic counterpoint to the rich structure theory of non-disintegrated minimal sets in $\mathrm{DCF}_0$ obtained in the 1990s (see the account in Jaoui 2021). For order-one autonomous equations the picture was settled classically (Rosenlicht 1974). Jaoui (2021) established the planar case: for every $d \ge 3$, the solution set of a general planar algebraic vector field of degree $d$ in a differentially closed field is strongly minimal and disintegrated. This was subsequently generalized: DeVilbiss and Freitag (2023) proved that a general order-$h$ equation with differentially transcendental coefficients is strongly minimal, and in a follow-up preprint (arXiv:2301.06362, posted January 2023, revised November 2025) Jaoui proved that a generic polynomial system on $\mathbb{A}^n$ is strongly minimal and geometrically trivial for every $n, d \ge 2$, with an analogous statement for generic vector fields on the complement of a smooth hyperplane section of a smooth projective variety of any dimension — so the degree-$2$ case, the higher-dimensional case, and the existence of explicit strongly minimal quadratic examples are all settled.

What remains unsettled is the strength of the algebraic independence between solutions. Write $(C_n)$ for the statement: if $\gamma_1, \ldots, \gamma_n$ are $n$ distinct non-stationary analytic solutions of the general vector field $v$ (a solution is stationary if it is constant, necessarily at one of the zeros of $v$; for a general field of degree $d$ there are $d^2$ of these singularities), then $\gamma_1, \ldots, \gamma_n$ are algebraically independent over $\mathbb{C}$. The case $n = 1$ is the classical transcendence result of Landis and Petrovskii for general planar systems (as recounted in Jaoui 2021). Jaoui's disintegration theorem gives the implication $(C_2) \Rightarrow (C_n)$ for all $n \ge 2$, and Freitag–Jaoui–Moosa (2022) showed more generally that for a type over the constants, independence of any two realizations already yields minimality and complete disintegration. So the single missing input is the binary statement $(C_2)$ itself. Jaoui conjectured the corresponding complete classification: for a general planar algebraic vector field of sufficiently large degree, the structure of the solution set is an expansion of a strictly disintegrated set (a pure infinite set with no additional structure) by $d^2$ constant symbols naming the singularities. For a strongly minimal solution set, a nontrivial binary algebraic relation is necessarily a generically finite-to-finite self-correspondence (an algebraic correspondence on the solution set commuting with the flow): any proper invariant subvariety of $X \times X$ projecting dominantly to both factors has finite generic fibers, since fibers over a solution are definable subsets of a strongly minimal set and hence finite or cofinite. Thus $(C_2)$ is equivalent to trivial internal structure — the absence of non-diagonal self-correspondences — which is precisely property (4) left open in Jaoui's November 2025 problem list (arXiv:2301.06362, Question 1.5.2) for the families of generic systems. For specific quadratic systems such as the classical Lotka–Volterra system, the analogous full independence is known (Duan–Nagloo 2026; Duan–Eagles–Jimenez 2025), but their proofs use special structure — classified invariant algebraic curves and invariant volume forms — that a general vector field does not possess.

Problem Statement

Prove or refute the statement $(C_2)$ of Jaoui's conjecture on general planar algebraic vector fields: there exists $d_0$ such that for every $d \ge d_0$ (the natural strong form is all $d \ge 3$) and every planar algebraic vector field $v$ of degree $d$ with $\mathbb{Q}$-algebraically independent coefficients, any two distinct non-stationary analytic solutions $\gamma_1, \gamma_2 : D \to \mathbb{A}^2(\mathbb{C})$ of the system $\frac{dx}{dt} = f(x,y)$, $\frac{dy}{dt} = g(x,y)$ — where $D \subset \mathbb{C}$ is a nonempty connected open domain — are algebraically independent over $\mathbb{C}$, that is

$$\operatorname{trdeg}_{\mathbb{C}} \mathbb{C}(\gamma_1, \gamma_2) = 4,$$

where $\mathbb{C}(\gamma_1, \gamma_2)$ is the field generated over $\mathbb{C}$ by the four coordinate functions of $\gamma_1$ and $\gamma_2$. Under the strong minimality already proved for $d \ge 3$, the statement is equivalent to each of the following formulations: the structure induced on the solution set of a general planar algebraic vector field of sufficiently large degree is exactly the expansion of a strictly disintegrated set (a pure infinite set with no additional structure) by $d^2$ constant symbols naming the $d^2$ singularities of $v$; and, for generic parameters, every generically finite-to-finite algebraic self-correspondence of the solution set commuting with the flow is the diagonal (trivial internal structure, property (4) of Jaoui's Question 1.5.2). A positive answer, combined with the disintegration theorem already proved for $d \ge 3$, implies $(C_n)$ for all $n \ge 1$ and completes the classification of all algebraic relations among the solutions. A negative answer must exhibit, for arbitrarily large degrees $d$, a general vector field $v$ of degree $d$ — one whose coefficients avoid the exceptional countable union of proper $\mathbb{Q}$-defined Zariski-closed subsets of the coefficient space — together with two distinct non-stationary analytic solutions violating the displayed transcendence degree, equivalently a non-diagonal generically finite-to-finite self-correspondence of the solution set of $v$ commuting with the flow.

References

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