The Jacobi bound conjecture, originating in Jacobi's 1865 posthumous paper and made precise by Ritt in 1935, predicts that the order of a system of nn polynomial ordinary differential equations in nn unknowns — the number of arbitrary constants in a generic zero-dimensional component of its solution set — is bounded by the Jacobi number of the system, the optimal-assignment value of the matrix of orders of the equations in the unknowns. Li and Li (2016) settled the case where the prime differential ideal is presented by a characteristic set, proving the optimal bound ord⁡(I)≤Jac⁡(A)\operatorname{ord}(\mathcal{I})\leq\operatorname{Jac}(\mathcal{A}) and building on it single-exponential algorithms for differential Chow forms; the same conjecture for prime differential ideals presented by plain generator systems has remained the open core. A March 2026 preprint by Dupuy and Zureick-Brown proves the conjecture for all generically reduced components — covering most generator presentations, including every presentation whose generated differential ideal is itself prime, and reproving the characteristic-set case with equality under orderly rankings; it does not cover the non-generically-reduced case.

Background

Differential algebra studies systems of polynomial differential equations by algebraic means. Fix a differential field KK of characteristic zero with a single derivation ∂\partial (the ordinary case) and let K{y1,…,yn}K\{y_1,\dots,y_n\} be the ring of differential polynomials: ordinary polynomials in the differential unknowns yiy_i and their formal derivatives yi(r)y_i^{(r)}, with ∂(yi(r))=yi(r+1)\partial(y_i^{(r)})=y_i^{(r+1)}. A differential ideal is an ideal closed under ∂\partial; for differential polynomials u1,…,ueu_1,\dots,u_e, the notation [u1,…,ue][u_1,\dots,u_e] stands for the smallest differential ideal containing them, generated by the uiu_i and all their derivatives. A prime differential ideal I\mathcal{I} defines an irreducible differential variety. Its differential dimension and order are read off from the Kolchin differential dimension polynomial — the unique numerical polynomial ωI(t)\omega_{\mathcal{I}}(t) that equals, for all large tt, the transcendence degree over KK of the field generated by the derivatives up to order tt of a generic solution; it has the form ωI(t)=d(t+1)+h\omega_{\mathcal{I}}(t)=d(t+1)+h, where dd is the differential dimension and h=ord⁡(I)h=\operatorname{ord}(\mathcal{I}) is called the order of I\mathcal{I}. When d=0d=0, ord⁡(I)\operatorname{ord}(\mathcal{I}) is exactly the number of arbitrary constants (initial conditions) parametrizing a generic solution — the quantity Jacobi called the order of the system.

For a differential polynomial ff, let ord⁡yj(f)\operatorname{ord}_{y_j}(f) be the largest rr such that yj(r)y_j^{(r)} occurs effectively in ff (taken as −∞-\infty when yjy_j does not occur). The Jacobi number of a system u1,…,unu_1,\dots,u_n in nn differential unknowns is the optimal-assignment value of its order matrix, J(u1,…,un)=max⁡σ∈Sn∑i=1nord⁡yi(uσ(i)),J(u_1,\dots,u_n)=\max_{\sigma\in S_n}\sum_{i=1}^{n}\operatorname{ord}_{y_i}\big(u_{\sigma(i)}\big), that is, the best total order obtained by matching each equation to a distinct unknown.

Jacobi, in a posthumous 1865 paper edited by Borchardt, asserted that the number of parameters in the general solution of a differential system is at most its Jacobi number — a differential analogue of Bézout's degree bound. Ritt (1935) observed that Jacobi's notion of dimension had no rigorous definition, supplied one via transcendence degrees, declared the resulting statement an open problem, and proved the linear and two-variable cases together with a weaker general bound (the sum over the unknowns of the maximal order of occurrence in each unknown). The sharp statement has been open ever since. A century of partial results includes: Lando's first-order cases; Tomasovic's version for partial differential equations in low order; the quasi-regular case — systems whose linearization at a generic point has full rank — treated by Ollivier and Sadik and extended by Ollivier; and the case of systems independent over a prime differential ideal studied by Kondratieva, Mikhalev, and Pankratiev, who also survey the history. In the effective direction, general (much weaker) order bounds in terms of generators follow from work on the differential Nullstellensatz — the differential analogue of Hilbert's Nullstellensatz, characterizing the differential polynomials that vanish on all solutions of a differential system — by Golubitsky, Kondratieva, Ovchinnikov, and Szanto, from prolongation bounds for differential kernels (field extensions equipped with derivations defined on specified finite sets of elements, modeling finite-order truncations of solutions) by Gustavson and León Sánchez, and from proof-mining bounds (explicit effective bounds extracted from nonconstructive proofs by proof-theoretic methods) by Simmons and Towsner — none of these reaches the sharp Jacobi bound.

A key tool in the subject is the characteristic set: a ranking is a total ordering of the derivatives yi(r)y_i^{(r)} compatible with differentiation, and Ritt's division algorithm extracts from a prime differential ideal a canonical triangular finite subset A\mathcal{A} (its characteristic set) whose leading derivatives are distinct; an orderly ranking is one that orders derivatives primarily by order. For a prime differential ideal I=sat⁡(A)\mathcal{I}=\operatorname{sat}(\mathcal{A}) given by a characteristic set A\mathcal{A} under an arbitrary ranking — where sat⁡(A)\operatorname{sat}(\mathcal{A}) denotes the saturated ideal of A\mathcal{A}, the prime differential ideal of all differential polynomials that Ritt's reduction by A\mathcal{A} brings to zero, so that A\mathcal{A} is a characteristic set of sat⁡(A)\operatorname{sat}(\mathcal{A}) — Li and Li proved ord⁡(I)≤Jac⁡(A)\operatorname{ord}(\mathcal{I})\leq\operatorname{Jac}(\mathcal{A}), with equality under orderly rankings, thereby establishing the Jacobi bound conjecture in the characteristic-set presentation; this bound is the foundation of their single-exponential-time algorithms for the differential Chow form (the differential analogue of the classical Chow, or Cayley, form of an algebraic variety, which encodes a variety through its intersections with generic linear subspaces). The unsettled form of the conjecture concerns prime differential ideals presented by plain generator systems {u1,…,un}\{u_1,\dots,u_n\}: the Jacobi number of the generators can be much smaller than that of any characteristic set computed from them, and no technique in the published literature bridges the two. In March 2026, Dupuy and Zureick-Brown posted a preprint proving the conjecture for all generically reduced components — components whose local ring at the generic point is reduced, i.e. free of nilpotent infinitesimal structure; the non-generically-reduced case remains unresolved.

Problem Statement

Let KK be a differential field of characteristic zero with one derivation, let R=K{y1,…,yn}R=K\{y_1,\dots,y_n\} be the ring of differential polynomials in nn differential unknowns, and let u1,…,un∈Ru_1,\dots,u_n\in R be a system of nn differential polynomials; write [u1,…,un][u_1,\dots,u_n] for the differential ideal they generate (the uiu_i together with all their derivatives). For each uiu_i and unknown yjy_j, let ord⁡yj(ui)\operatorname{ord}_{y_j}(u_i) be the largest rr such that yj(r)y_j^{(r)} occurs effectively in uiu_i, taken as −∞-\infty when yjy_j does not occur in uiu_i. The Jacobi number of the system is the optimal-assignment value of its order matrix, J(u1,…,un)=max⁡σ∈Sn∑i=1nord⁡yσ(i)(ui),J(u_1,\dots,u_n)=\max_{\sigma\in S_n}\sum_{i=1}^{n}\operatorname{ord}_{y_{\sigma(i)}}(u_i), that is, the largest total order obtainable by matching the nn equations to the nn unknowns injectively. Let I\mathcal{I} be a prime differential ideal minimal over [u1,…,un][u_1,\dots,u_n] and of differential dimension zero — meaning that the field generated by the derivatives of a generic solution has finite transcendence degree over KK (no free functions, only finitely many constants) — so that its order ord⁡(I)\operatorname{ord}(\mathcal{I}), the constant term of its Kolchin differential dimension polynomial (the numerical polynomial measuring the transcendence degree over KK of the derivatives of a generic solution up to order tt, for large tt), equivalently the number of arbitrary constants in a generic solution, is finite. The Jacobi bound conjecture, in the generator-presentation form left open by the source paper, asserts: ord⁡(I)≤J(u1,…,un)\operatorname{ord}(\mathcal{I})\leq J(u_1,\dots,u_n) for every such system over every differential field of characteristic zero and every such zero-dimensional prime component. Equivalently, in geometric language: every irreducible component of finite Krull dimension of the differential scheme cut out by u1,…,unu_1,\dots,u_n has Krull dimension at most J(u1,…,un)J(u_1,\dots,u_n) (the Krull dimension of a finite-dimensional component being exactly its order). The conjecture is settled either by a general proof of the inequality, or by a concrete system together with a zero-dimensional prime component whose order exceeds the system's Jacobi number. The characteristic-set-presented case is already settled, with equality under orderly rankings, so the open content is genuinely about presentations by generators; the bound, if true, is optimal.

References

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