Abhyankar–Sathaye embedding conjecture for hypersurfaces in affine space

The Abhyankar–Sathaye embedding conjecture predicts that, over an algebraically closed field of characteristic zero (the standard setting of the conjecture; some authors pose it over every field of characteristic zero), an irreducible polynomial $f$ in $n$ variables whose zero hypersurface in affine $n$-space is isomorphic to affine $(n-1)$-space must be a coordinate (a member of a polynomial generating system, equivalently the image of a variable under a polynomial automorphism of $\mathbb A^n$); equivalently, every closed embedding $\mathbb A^{n-1}\hookrightarrow\mathbb A^n$ is rectifiable by an automorphism of the ambient space. The case $n\leqslant 2$ is the classical theorem of Abhyankar–Moh and Suzuki; the conjecture is open for every $n\geqslant 3$, starting with embeddings of $\mathbb A^2$ in $\mathbb A^3$.

Background

Affine algebraic geometry studies polynomial maps between affine spaces. The affine $n$-space $\mathbb A^n_K$ over a field $K$ is the variety with coordinate ring $K[x_1,\dots,x_n]$; a polynomial automorphism of $\mathbb A^n$ is a polynomial map $\sigma=(\sigma_1,\dots,\sigma_n)$ with a polynomial inverse, and a polynomial $f\in K[x_1,\dots,x_n]$ is called a coordinate or variable if $f=\sigma(x_1)$ for such a $\sigma$, equivalently if there exist $g_2,\dots,g_n$ with $K[f,g_2,\dots,g_n]=K[x_1,\dots,x_n]$. Geometrically, the zero locus $V(f)$ of a coordinate is the image of a coordinate hyperplane under an automorphism.

The embedding conjecture of Abhyankar and Sathaye, formulated in their 1970s work on algebraic space curves, asks the converse: over an algebraically closed field of characteristic zero, if an irreducible polynomial defines a hypersurface isomorphic to $\mathbb A^{n-1}$, must the polynomial be a coordinate? Equivalently, every closed embedding $\mathbb A^{n-1}\hookrightarrow\mathbb A^n$ is rectifiable, i.e. becomes the standard inclusion after composition with an automorphism of the ambient space. For $n=2$ the answer is the classical theorem of Abhyankar–Moh and Suzuki: an irreducible plane curve isomorphic to $\mathbb A^1$ is sent to a coordinate line by an automorphism of the plane. Their proof introduced approximate roots — for a monic polynomial $f$ of degree $d$ in $y$, polynomials $g$ with $f=g^j+(\text{lower }y\text{-degree})$ for $j\mid d$, generalizing square and cube roots — together with the semigroup of intersection multiplicities at infinity, and showed the curve is a coordinate line exactly when this semigroup is generated by the degree.

Two facts make the case $n\geqslant 3$ far harder. First, the automorphism group of the affine plane has a structure theorem (Jung–van der Kulk: every plane automorphism is a composition of affine and triangular ones), which yields effective procedures in dimension two; for $n\geqslant 3$ there exist wild automorphisms, such as Nagata's famous example in three variables, and no normal form or effective criterion for recognizing coordinates is known. Second, the standard algebraic tool is the locally nilpotent derivation: a $K$-linear derivation $D$ of $K[x_1,\dots,x_n]$ (a map satisfying the Leibniz rule) with $D^m=0$ for $m$ large; a polynomial $f$ is a variable exactly when some locally nilpotent derivation satisfies $D(f)=1$, and deciding the non-existence of such a derivation for a concrete polynomial in $n\geqslant 3$ variables is an open-ended task.

Assi's work [4] is set in the quasi-ordinary world. A polynomial $f(\underline x,y)$ monic in $y$ over $\mathbf K[\underline x]$, $\underline x=(x_1,\dots,x_e)$, is quasi-ordinary (a class going back to Jung and Abhyankar) when its discriminant in $y$ is a monomial in $\underline x$ times a unit; such polynomials behave combinatorially like plane curve branches and carry a discrete value semigroup. Assi (J. Pure Appl. Algebra 214 (2010)) extended the Abhyankar–Moh theory of approximate roots and semigroups to quasi-ordinary polynomials, established stability of irreducibility, approximate roots and the semigroup under one-parameter translations $F\mapsto F-\lambda$, and proved the embedding conjecture for polynomials whose associated meromorphic polynomial is quasi-ordinary, also after a change of coordinates (the 'almost quasi-ordinary' case). His paper states that the conjecture remains open outside these classes, in fact for ambient dimension greater than $2$, and records further open sub-questions: effective recognition of almost quasi-ordinary polynomials for $e\geqslant 3$ (blocked by the unknown structure of the automorphism group of $\mathbb A^e$), the behavior of the quasi-ordinary structure under deformations beyond translations, and the characterization of the coordinate changes that produce the quasi-ordinary property.

The unrestricted conjecture is a central open problem. Kaliman proved that a polynomial in $\mathbb C[x,y,z]$ whose general fiber is $\mathbb C^2$ is a variable, settling the 'hyperplane' version (all fibers affine planes) in ambient dimension $3$, but the conjecture hypothesizes only that the single fiber $f=0$ is an affine space. Kishimoto settled the three-dimensional problem under restrictions on the boundary at infinity; Ohta settled a geometric case in all dimensions via the canonical compactification; Popov proved a partial case for embeddings equivariant under unipotent group actions. On the negative side, the Vénéreau polynomials — four-variable hyperplanes, i.e. polynomials all of whose fibers are $\mathbb A^3$ — are not known to be coordinates and stand as potential counterexamples, while earlier proposed counterexamples of Shpilrain and Yu were refuted by van den Essen and van Rossum. Since 2022 the Gupta school has proved the conjecture for widening families of linear hypersurfaces $\alpha(X)Y-F(X,Z,T)$ and connected its rectifiability question to the Zariski cancellation problem (whether $X\times\mathbb A^m\cong Y\times\mathbb A^m$ forces $X\cong Y$) and to $\mathbb A^1$-contractibility of associated varieties (contractibility to a point in the $\mathbb A^1$-homotopy sense).

Problem Statement

Conjecture (Abhyankar–Sathaye, codimension-one embedding problem). Let $K$ be an algebraically closed field of characteristic zero and let $n\geqslant 3$. If $f\in K[x_1,\dots,x_n]$ is irreducible and its zero hypersurface $V(f)\subset\mathbb A^n_K$ is isomorphic to $\mathbb A^{n-1}_K$, then $f$ is a coordinate of $K[x_1,\dots,x_n]$: there exist $g_2,\dots,g_n$ with $K[f,g_2,\dots,g_n]=K[x_1,\dots,x_n]$, equivalently $f=\sigma(x_1)$ for some polynomial automorphism $\sigma$ of $\mathbb A^n_K$. Equivalently: every closed embedding $\mathbb A^{n-1}\hookrightarrow\mathbb A^n$ over $K$ is rectifiable, i.e. becomes the standard inclusion after composition with an automorphism of $\mathbb A^n$. This is the standard formulation over algebraically closed fields; some authors pose the conjecture over every field of characteristic zero.

Prove the conjecture for all $n\geqslant 3$ — either for every algebraically closed field of characteristic zero (which resolves the conjecture in its standard form), or for every field of characteristic zero (the stronger variant) — or refute it by exhibiting, for some $n\geqslant 3$ and some algebraically closed field $K$ of characteristic zero, an irreducible $f\in K[x_1,\dots,x_n]$ with an isomorphism $V(f)\cong\mathbb A^{n-1}_K$, such that $f$ is provably not a coordinate of $K[x_1,\dots,x_n]$. The case $n\leqslant 2$ is the theorem of Abhyankar–Moh and Suzuki and is not part of the open problem. The target is the unrestricted statement: known positive results impose additional hypotheses — quasi-ordinariness (Assi [4]), conditions on the general fiber (Kaliman), on the boundary at infinity (Kishimoto), on the canonical compactification (Ohta), or equivariance (Popov) — and partial progress such as a proof for $n=3$ alone or for further special classes of polynomials falls short of a resolution in either direction.

References

1. S. S. Abhyankar and T. T. Moh, 'Embeddings of the line in the plane', Journal für die reine und angewandte Mathematik 276 (1975), 148–166. DOI: 10.1515/crll.1975.276.148, https://doi.org/10.1515/crll.1975.276.148
2. M. Suzuki, 'Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace C^2', Journal of the Mathematical Society of Japan 26 (1974), no. 2, 241–257. DOI: 10.2969/jmsj/02620241, https://doi.org/10.2969/jmsj/02620241
3. S. S. Abhyankar and A. M. Sathaye, 'Geometric Theory of Algebraic Space Curves', Lecture Notes in Mathematics, vol. 423, Springer (1974). DOI: 10.1007/bfb0066166, https://doi.org/10.1007/bfb0066166
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