Completeness of projective differential algebraic varieties: classification, the zero-dimensional test reduction, and the constant Kolchin polynomial question

In differential algebraic geometry, a differential variety $V\subseteq\mathbb{P}^n$ is complete when for every quasiprojective differential variety $Y$ the projection $V\times Y\to Y$ is closed in the Kolchin topology. Building on Kolchin, Pong, and Freitag, the paper of Freitag, León Sánchez, and Simmons (Communications in Algebra, 2016) proved that completeness is exactly what lets Kolchin's linear dependence theory extend from projective varieties in the constants to general differential varieties, and posed the classification problem: which projective differential varieties are complete? Every complete differential variety is zero-dimensional, but Simmons' first zero-dimensional incomplete projective example shows the classification is genuinely open. The source-faithful open problem cluster comprises: the classification question (Question 3.6), the conjectured reduction of completeness tests to zero-dimensional test varieties (Conjecture 5.3), which follows from the Weak Catenary Conjecture (Conjecture 5.2), and whether every complete differential variety has constant Kolchin polynomial in the case of several derivations (Remark 4.7). None of these components has been resolved in the literature to date.

Background

Work in characteristic zero with a differential field $(k,\Delta)$: a field equipped with finitely many commuting derivations $\delta_1,\dots,\delta_m$ ($m=1$ is the ordinary case, $m>1$ the partial case). Systems of differential polynomial equations, i.e. polynomial equations in the variables and their iterated derivatives, cut out differential algebraic varieties, which carry the Kolchin topology, whose closed sets are solution sets of such systems; it refines the Zariski topology. The natural universe is a differentially closed field $\mathcal{U}$ (a model of the theory $DCF_{0,m}$), the differential analogue of an algebraically closed field.

For algebraic varieties, completeness — the property of a projective variety $X$ that $X\times Y\to Y$ is a closed map for every variety $Y$, equivalent to properness — has a direct differential analogue: a $\Delta$-closed $V\subseteq\mathbb{P}^n$ is complete if $\pi_2:V\times Y\to Y$ is a closed map of Kolchin topologies for every quasiprojective differential variety $Y$. Kolchin introduced the notion in the 1970s: the constant points of a projective variety are complete, $\mathbb{P}^n$ itself is not, and he used completeness to extend the classical Wronskian criterion (linear dependence of meromorphic functions over the constants is detected by the vanishing of a determinant of derivatives) to linear dependence over an arbitrary projective variety in the constants. Pong (2000) developed the theory over differentially closed fields in one derivation via a differential version of the valuative criterion for properness and proved that every complete differential variety is zero-dimensional (zero differential dimension, i.e. finite rank). Freitag (2014) extended the valuative criterion and the zero-dimensionality theorem to the partial case and produced new classes of examples. Simmons (2013 thesis; 2015/2017 paper) constructed the first example of a zero-dimensional projective differential variety that is not complete — in one derivation, the $\delta$-closed subset of $\mathbb{P}^1\times\mathbb{A}^1$ cut out by $x''=x^3$ together with $2yx^4-4y(x')^2=1$, whose second projection has the non-closed image $\{y:y'=0,\ y\neq 0\}$ — so zero-dimensionality alone does not characterize completeness.

Freitag, León Sánchez, and Simmons (2016) connected completeness to linear dependence: a tuple $(a_0,\dots,a_n)$ is linearly dependent over a projective differential variety $V$ if $\sum v_i a_i=0$ for some point $[v_0:\dots:v_n]\in V$; the set $ld(V)$ of such tuples is always a differential variety, and when $V$ is complete, membership in $ld(V)$ is characterized by the vanishing of a system of differential polynomial equations, extending Kolchin's theorem. The Kolchin polynomial $\omega_V(t)$ of an irreducible differential variety — the differential analogue of the Hilbert polynomial, a numerical polynomial that for large $t$ measures the transcendence degree gained by adjoining all derivatives of a generic point up to order $t$; its degree is the differential type of $V$ — controls the structure: if a complete $V$ has constant Kolchin polynomial equal to $d$, then $\omega_{ld(V)}(t)=(n-1)\binom{m+t}{m}+d$ (Proposition 4.6), and in the ordinary case there is a unique (up to a nonzero factor) irreducible differential polynomial $R$ of order $d$ whose general solution is exactly the set of linearly dependent tuples over $V$ (Corollary 4.8). In the ordinary case the constant-Kolchin-polynomial hypothesis is automatic, because every complete variety there has differential type zero; in the partial case it is unknown whether every complete differential variety has constant Kolchin polynomial at all.

The obstruction to a classification is structural: verifying completeness a priori quantifies over all test varieties $Y$, and the differential valuative criteria are difficult to apply to nonlinear equations beyond order one. The paper identifies a conjectural reduction: since complete varieties are zero-dimensional, one hopes that only zero-dimensional $Y$ need be tested (Conjecture 5.3), which would follow from the Weak Catenary Conjecture: every positive-dimensional irreducible differential variety $V$ contains a proper irreducible differential subvariety meeting a given zero-dimensional differential subvariety $W\subseteq V$ without being contained in it (Conjecture 5.2). This is a weak form of the long-standing Kolchin catenary problem, which asks whether the lattice of irreducible differential subvarieties admits maximal chains of the length dictated by the Kolchin polynomial (maximal chains of irreducible sets whose Kolchin polynomials decrease strictly at each step, so that all maximal chains between comparable sets have the same length); Pong (2002) settled the catenary problem only when $V$ is an algebraic variety, using resolution of singularities, which is unavailable in differential algebraic geometry. The same paper also poses stronger catenary-type questions (Questions 5.13 and 5.14) about openness of the maps induced on differential spectra (the prime spectrum of a differential ring, equipped with the topology whose closed sets correspond to differential ideals) and on prolongation spaces — the differential-algebraic analogues of arc spaces — by finite open algebraic maps, and notes that even the effective computation of the polynomial $R$ requires hard differential elimination. These adjacent questions are peripheral to the core classification problem.

Problem Statement

Let $\mathcal{U}$ be a differentially closed field of characteristic zero with $m$ commuting derivations $\Delta=\{\delta_1,\dots,\delta_m\}$, and let $K\subset\mathcal{U}$ be a differential subfield. A $\Delta$-closed set $V\subseteq\mathbb{P}^n$ defined over $K$ is complete if for every quasiprojective differential variety $Y$ the projection $\pi_2:V\times Y\to Y$ is a closed map for the Kolchin topology.

The problem, posed as Question 3.6 of Freitag–León Sánchez–Simmons, is to classify the complete projective differential varieties: give a characterization that is necessary and sufficient for completeness. Every complete differential variety is zero-dimensional, and Simmons' example shows that not every zero-dimensional projective differential variety is complete, so the classification must make a genuine distinction inside the zero-dimensional class.

The problem includes, as explicitly linked components identified by the source:

1. The zero-dimensional test reduction (Conjecture 5.3): prove or refute that a $\Delta$-closed $V\subseteq\mathbb{P}^n$ is complete if and only if $\pi_2:V\times Y\to Y$ is closed for every quasiprojective zero-dimensional differential variety $Y$.

2. The Weak Catenary Conjecture (Conjecture 5.2), which implies component 1: for every positive-dimensional irreducible differential variety $V\subseteq\mathbb{A}^n$ and every zero-dimensional differential subvariety $W\subseteq V$, there is a proper irreducible differential subvariety $V_1\subsetneq V$ with $V_1\cap W\neq\emptyset$ and $V_1\nsubseteq W$.

3. The Kolchin polynomial structure question (Remark 4.7): in the partial case $m>1$, determine whether every complete differential variety has constant Kolchin polynomial — a fact automatic for $m=1$ — since this is the hypothesis under which the linear dependence theorem over $V$ (existence and uniqueness of the characterizing differential polynomial $R$ of Corollary 4.8) is presently known.

These components jointly support one objective, an effective structure theory for complete differential varieties: components 1 and 2 reduce the a priori unbounded quantification over test varieties in the definition of completeness, component 3 supplies the invariant that controls the linear-dependence theory which completeness exists to serve, and component 2 (the Weak Catenary Conjecture) implies component 1 (the zero-dimensional test reduction).

References

1. James Freitag, Omar León Sánchez, and William Simmons, "On Linear Dependence Over Complete Differential Algebraic Varieties", Communications in Algebra 44(6):2645–2669 (2016). DOI: 10.1080/00927872.2015.1057828; arXiv:1401.6211. URL: https://arxiv.org/abs/1401.6211
2. William D. Simmons, "New examples (and counterexamples) of complete finite-rank differential varieties", Communications in Algebra 45(7):3137–3149 (2017). DOI: 10.1080/00927872.2016.1236115; arXiv:1505.03430. URL: https://arxiv.org/abs/1505.03430
3. James Freitag, "Completeness in partial differential algebraic geometry", Journal of Algebra 420:350–372 (2014). DOI: 10.1016/j.jalgebra.2014.07.025; arXiv version titled "Completeness in Partial Differential Fields": arXiv:1106.0703. URL: https://arxiv.org/abs/1106.0703
4. Wai Yan Pong, "Complete Sets in Differentially Closed Fields", Journal of Algebra 224(2):454–466 (2000). DOI: 10.1006/jabr.1999.8077. URL: https://doi.org/10.1006/jabr.1999.8077
5. Wai Yan Pong, "Chains of Differential Subvarieties in an Algebraic Variety", Journal of Algebra 248(2):755–764 (2002). DOI: 10.1006/jabr.2001.9065. URL: https://doi.org/10.1006/jabr.2001.9065
6. E. R. Kolchin, "Differential Equations in a Projective Space and Linear Dependence over a Projective Variety", in Contributions to Analysis (Academic Press, 1974), pp. 195–214. DOI: 10.1016/b978-0-12-044850-0.50024-1. URL: https://doi.org/10.1016/b978-0-12-044850-0.50024-1
7. James Freitag, Omar León Sánchez, Wei Li, and Joel Nagloo, with an appendix by Michael F. Singer, "On the number of independent solutions of algebraic differential equations" (2026). arXiv:2603.12387. URL: https://arxiv.org/abs/2603.12387