Commuting Locally Nilpotent Derivations Conjecture

Definitions and mathematical statement

Commuting derivations conjecture $\mathrm{CD}(n)$. Let $A={\mathbb{C}}[x_1,\ldots,x_n]$ and let $D_1,\ldots,D_{n-1}$ be pairwise commuting locally nilpotent ${\mathbb{C}}$-derivations, linearly independent over $A$. Must there be a coordinate $f\in A$ such that\[\bigcap_{i=1}^{n-1}\ker D_i={\mathbb{C}}[f]?\] The nontrivial range is $n\ge2$; for $n=1$, with the empty-intersection convention, the assertion simply concerns $A={\mathbb{C}}[x_1]$. The source does not explicitly state a dimension range.

A derivation is ${\mathbb{C}}$-linear and satisfies $D(ab)=aD(b)+D(a)b$. Locally nilpotent means that for every $a\in A$ there is an integer $m\ge1$, allowed to depend on $a$, with $D^m(a)=0$. Pairwise commutation means $D_iD_j=D_jD_i$. Independence over $A$ means that $\sum_i a_iD_i=0$ as a derivation, with $a_i\in A$, forces all $a_i=0$. It is stronger than independence over ${\mathbb{C}}$, but does not assert pointwise independence of the resulting vector fields at every point of affine space.

A coordinate means that $f$ extends to a polynomial coordinate system: there are $f_2,\ldots,f_n\in A$ with $A={\mathbb{C}}[f,f_2,\ldots,f_n]$. The requested conclusion is therefore stronger than saying the common kernel is abstractly a polynomial ring in one variable, or is generated by one arbitrary polynomial. No common slice, everywhere-free action, or stronger independence condition is inserted.

The supplied triage checked 2026-08-21 labels the conjecture open and reports no located complete resolution, with the report as its sole source. It provides no precise dimension-specific theorem or later comparison. This dated assertion is retained without independent verification of present-day status.

Short English statement

For $n-1$ commuting locally nilpotent derivations independent over the polynomial ring, ask whether their joint invariants are generated by a polynomial coordinate. Ring-linear independence is not everywhere pointwise independence, and an arbitrary one-variable invariant ring would be a weaker conclusion.

Sources

• Local UnsolvedMath registry, permanent ID 30000653, code catalogue entry; Commuting Locally Nilpotent Derivations Conjecture, labeled commuting derivations conjecture $\mathrm{CD}(n)$ in the original statement; Oberwolfach Reports Open Problems.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/01, pp. 5--82. https://doi.org/10.4171/owr/2007/01. Proposed year 2007; no proposer named.
• The same full report is the sole supplied literature source; no separate authoritative coordinate or derivation theorem is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field records label removal and notation normalization. Verify dated claims independently.