Determine the automorphism group of the rank-$n$ Witt algebra $W_n=\mathrm{Der}\,\mathbb{C}[x_1,\dots,x_n]$ (equivalently the polynomial automorphism group $\mathrm{Aut}\,\mathbb{C}[x_1,\dots,x_n]$) for $n\ge 3$

For the polynomial algebra $A_n=\mathbb{C}[x_1,\dots,x_n]$ and the rank-$n$ Witt algebra $W_n=\mathrm{Der}\,A_n$ (the Lie algebra of polynomial vector fields), Han and Su (2015) established a group isomorphism $\mathrm{Aut}\,A_n\cong\mathrm{Aut}\,W_n$ for every $n$, computed $\mathrm{Aut}\,W_2$, and proved that in each rank $n$ the Jacobian conjecture is equivalent to the conjecture that every nonzero endomorphism of $W_n$ is an automorphism. The endomorphism-conjecture strand was refuted in July 2026: an explicit three-variable polynomial map with constant nonzero Jacobian determinant and no polynomial inverse was announced and mechanically verified, and counterexamples now exist in every dimension above two. The surviving open core is the source paper's title problem: give a complete description of $\mathrm{Aut}\,W_n\cong\mathrm{Aut}\,A_n$ for $n\ge 3$, open even for $\mathbb{C}[x,y,z]$, where wild (non-tame) automorphisms are known to exist.

Background

Let $A_n=\mathbb{C}[x_1,\dots,x_n]$ be the polynomial algebra in $n$ variables. Its algebra automorphisms are the polynomial maps of $\mathbb{C}^n$ with polynomial inverses. Write $\partial_i=\partial/\partial x_i$ and $W_n=\mathrm{Der}\,A_n$ for the Lie algebra of derivations of $A_n$, i.e. the polynomial vector fields $\sum_i p_i\partial_i$ with bracket $[D,E]=DE-ED$; $W_n$ is the rank-$n$ (one-sided) Witt algebra, the first known example of an infinite-dimensional simple Lie algebra. It must be distinguished from the two-sided Witt algebra $\mathrm{Der}\,\mathbb{C}[x_1^{\pm1},\dots,x_n^{\pm1}]$ of the Laurent polynomial algebra, whose automorphism group was determined earlier by other methods; for $W_n$ the standard foothold for automorphism arguments — the set of ad-locally-finite elements (elements $x$ whose iterated brackets $\mathrm{ad}_x^k$ span a finite-dimensional space) — is inaccessible, which is what makes $\mathrm{Aut}\,W_n$ hard, open for every $n\ge 2$ before 2015. A Jacobi tuple is an $n$-tuple of polynomials $(f_1,\dots,f_n)$ whose Jacobian determinant $J(f_1,\dots,f_n)=\det(\partial f_i/\partial x_j)$ is a nonzero constant; a polynomial map with this property is called a Keller map. Keller's Jacobian conjecture (1939) asserts that every Keller map of $\mathbb{C}^n$ is invertible with polynomial inverse, i.e. that every Jacobi tuple generates $A_n$. Zhao (1998) posed the companion Witt algebra conjecture — every nonzero endomorphism of the Lie algebra $W_n$ is an automorphism (equivalently, is surjective, since such endomorphisms are automatically injective) — and proved that it implies the Jacobian conjecture. On the automorphism side, an automorphism of $A_n$ is called tame if it is a finite composition of affine automorphisms ($x_i\mapsto\sum_j a_{ij}x_j+b_i$ with invertible matrix) and triangular automorphisms ($x_i\mapsto x_i+p(x_1,\dots,x_{i-1})$, fixing the other variables). For $n\le 2$ classical work from the 1940s–50s shows all automorphisms are tame, so $\mathrm{Aut}\,A_2$ is explicitly known; for $n\ge3$ the tame automorphisms do not exhaust the group: Shestakov and Umirbaev (2003–2004) proved that a famous 1972 example of Nagata in three variables is wild (not tame), so $\mathrm{Aut}\,A_n$ is strictly larger than its tame subgroup for $n\ge3$, and no complete description — generators, presentation, or structural characterization — is known in any rank $n\ge3$. Han and Su (2015) connected the two sides: they characterized the semigroup of nonzero endomorphisms of $W_n$ via Jacobi tuples, $\mathrm{End}\,W_n\setminus\{0\}\cong JT_n$, proved the Jacobian conjecture equivalent (in each rank $n$) to the Witt algebra conjecture, established the group isomorphism $\zeta\colon\mathrm{Aut}\,A_n\cong\mathrm{Aut}\,W_n$ (the paper's own notation, used below), and thereby computed $\mathrm{Aut}\,W_2$. In July 2026 the Jacobian conjecture was refuted in dimension three: an explicit Keller map $F:\mathbb{C}^3\to\mathbb{C}^3$ with $\det DF\equiv-2$ that is generically three-to-one was announced by Alpöge, an infinite family was given by Gallagher, expositions followed by Tao and Speyer, and Gao extended the construction to every dimension above two. Consequently the Witt algebra conjecture fails for $W_n$ for every $n\ge3$, and the isomorphisms of Han–Su turn each non-invertible Keller tuple into an explicit nonzero non-surjective endomorphism of $W_n$. The Jacobian and Witt conjectures remain open in dimension $n=2$. What the refutation does not do is describe the automorphism group: the determination of $\mathrm{Aut}\,A_n\cong\mathrm{Aut}\,W_n$ for $n\ge3$ survives untouched as the source paper's principal open problem.

Problem Statement

Give a complete description — an explicit generating set, a presentation, or a structural characterization — of the automorphism group $\mathrm{Aut}\,W_n$ of the rank-$n$ Witt algebra $W_n=\mathrm{Der}\,\mathbb{C}[x_1,\dots,x_n]$ for each $n\ge3$; by the Han–Su isomorphism $\zeta$ this is the same problem as describing the polynomial automorphism group $\mathrm{Aut}\,\mathbb{C}[x_1,\dots,x_n]$ for $n\ge3$ (the source paper works over $\mathbb{C}$; the same question is meaningful over any field of characteristic $0$). Since wild automorphisms such as the Nagata automorphism are known to exist (Shestakov–Umirbaev), the description cannot be exhausted by the affine and triangular (tame) generators and must capture the full non-tame group. The original generality is retained: all $n\ge3$ and the entire group, with no restriction to tame, stably tame, or Jacobian-preserving subfamilies.

References

1. Jianzhi Han, Yucai Su, Automorphism groups of Witt algebras, arXiv:1502.01441 (2015), DOI 10.48550/arXiv.1502.01441, https://arxiv.org/abs/1502.01441
2. Kaiming Zhao, Isomorphisms between generalized Cartan type W Lie algebras in characteristic 0, Canadian Journal of Mathematics 50 (1998), 210–224, DOI 10.4153/cjm-1998-011-6, https://doi.org/10.4153/cjm-1998-011-6
3. Ivan Shestakov, Ualbai Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, Journal of the American Mathematical Society 17 (2004), 197–227, DOI 10.1090/S0894-0347-03-00440-5, https://doi.org/10.1090/S0894-0347-03-00440-5
4. Ivan Shestakov, Ualbai Umirbaev, The Nagata automorphism is wild, Proceedings of the National Academy of Sciences 100 (2003), 12561–12563, DOI 10.1073/pnas.1735483100, https://doi.org/10.1073/pnas.1735483100
5. Ott-Heinrich Keller, Ganze Cremona-Transformationen, Monatshefte für Mathematik und Physik 47 (1939), 299–306, DOI 10.1007/BF01695502, https://doi.org/10.1007/BF01695502
6. Terence Tao, A digestion of the Jacobian conjecture counterexample, blog post 'What's new', 21 July 2026, https://terrytao.wordpress.com/2026-07-21/a-digestion-of-the-jacobian-conjecture-counterexample/
7. Alexis Gallagher, An infinite family of counterexamples to the Jacobian Conjecture in dimension three: every generic fiber degree $n\ge3$ occurs, preprint, Zenodo, 20 July 2026, DOI 10.5281/zenodo.21479195, https://zenodo.org/records/21479195
8. David E. Speyer, The geometry and structure of Gallagher's counterexamples to the Jacobian conjecture, expository note, 23 July 2026, https://sbseminar.wordpress.com/wp-content/uploads/2026/07/jacobiantangentsweep.pdf
9. Shuhong Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (2026), DOI 10.48550/arXiv.2608.00222, https://arxiv.org/abs/2608.00222
10. The new counterexample to the Jacobian conjecture, Secret Blogging Seminar (group blog), 20 July 2026, https://sbseminar.wordpress.com/2026-07-20/the-new-counterexample-to-the-jacobian-conjecture/
11. L. Alpöge, A counterexample to the Jacobian conjecture in dimension three, announcement of 19 July 2026 (post on X; cited as reference [11] of Gao, arXiv:2608.00222), https://x.com/__alpoge__/status/2079028340955197566