1 · The essentials
The manuscripts describe a polynomial in four complex variables that cannot serve as one of the variables in a reversible polynomial change of variables. Once a fifth variable is allowed, however, it can serve that role. This contradicts the Stable Coordinate conjecture in four variables. The example has another striking feature: for every fixed output value, the points giving that value form a space equivalent to ordinary three-dimensional complex space. Yet no reversible polynomial change of the four surrounding variables can put that space into the standard position obtained by fixing one variable. This also contradicts the Abhyankar–Sathaye embedding conjecture in dimension four. Both manuscripts are marked as Lean-formalized in the supplied source.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a familiar-shaped object positioned inside a container so that the permitted rearrangements cannot put it into a standard position. Adding an extra direction of movement makes the rearrangement possible. Here the permitted rearrangements are reversible polynomial changes, not physical movements.
Why it matters
The example separates two questions: what a space is like in itself, and how it sits inside a larger space. It also shows that making a polynomial into a coordinate after adding a variable does not guarantee that it was already a coordinate. The main manuscript combines these failures in one example whose every fiber is affine three-space.
2 · Going further
A polynomial is a coordinate if it can be included among the variables of a polynomial coordinate system: a polynomial change of variables with a polynomial inverse. The main manuscript constructs an explicit degree-five polynomial that is not a coordinate in this ring but becomes a coordinate in . This is the stated counterexample to the Stable Coordinate conjecture in four variables.
For each , the fiber is isomorphic to affine three-space . Nevertheless, its embedding in is not rectifiable: no polynomial automorphism of the surrounding space carries it to a standard coordinate hyperplane. The distinction is between an isomorphism describing the fiber itself and a change of coordinates in the entire surrounding space. The main manuscript therefore also supplies the stated counterexample to the Abhyankar–Sathaye embedding conjecture in ambient dimension four.
The construction strategy reported in the abstracts is to exhibit an explicit polynomial with these contrasting properties. The supplied text does not give its formula or the argument establishing noncoordinate behavior, stabilization, or the fiber descriptions. The companion manuscript focuses on an explicit noncoordinate polynomial whose zero fiber is affine three-space, and states that adjoining variables gives counterexamples in every ambient dimension at least four. It does not specify the relation between the two explicit polynomials or detail that higher-dimensional construction.
Useful notions
- Polynomials over the complex numbers
- Polynomial changes of variables
- Affine space and polynomial zero sets
- Isomorphisms and embeddings
What was known before / what is new
Before
The source identifies the Stable Coordinate conjecture and the Abhyankar–Sathaye embedding conjecture as the claims being contradicted. In the settings addressed here, they concern whether becoming a coordinate after adding variables, or having an affine-space zero fiber, forces coordinate behavior. No further historical background or earlier partial results are supplied.
New
The main manuscript gives an explicit degree-five example in four complex variables that becomes a coordinate after adjoining just one variable, while not being a coordinate originally. Every fiber is affine three-space, and none of those embeddings is rectifiable. The companion manuscript gives an explicit affine-three-space zero-fiber counterexample and states a construction of embedding counterexamples in all ambient dimensions at least four.
Limits and assumptions
The main example is over the complex numbers, has degree five, and starts in four variables; the source does not state analogous results over other fields or in smaller dimensions. Its stabilization uses a single additional variable. The companion's higher-dimensional claim concerns embedding counterexamples and should not be read as saying that the main polynomial stays noncoordinate after adjoining a variable. Both manuscripts are labeled Lean-formalized, so the main result has a Lean formalization, but the supplied text does not identify the individual formal statements or establish that every accompanying claim is covered. It also provides no list of remaining open cases.
Mini glossary
- Coordinate polynomial —
- A polynomial that can serve as one variable in a polynomial change of variables having a polynomial inverse.
- Stable coordinate —
- A polynomial that becomes a coordinate after additional variables are adjoined.
- Fiber —
- The set of points at which a polynomial takes a specified value.
- Affine three-space —
- The space of triples of complex numbers, viewed with its polynomial structure.
- Rectifiable embedding —
- An embedding that a polynomial automorphism of the surrounding affine space can carry to a standard coordinate position.
- Polynomial automorphism —
- An invertible polynomial change of variables whose inverse is also polynomial.
Keywords : Stable coordinates · Affine-space fibers · Abhyankar–Sathaye conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A stable coordinate that is not a coordinate in four variables
05/10/2026We construct an explicit degree-five polynomial over that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.
An explicit noncoordinate polynomial with affine three-space zero fibre
24/09/2026We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.