A stable-coordinate counterexample in four variables

The explanation

Take the polynomial x + y² in two variables. Replace x by x − y², and it becomes plain x. That replacement is a polynomial automorphism [a change of variables made of polynomials that other polynomials can undo]. Here the undo step is x → x + y². Call a polynomial a coordinate if some such change turns it into one variable. A coordinate is one axis in disguise.

Now take four complex variables, x₁ to x₄. First set Q = x₂² − x₄² + x₁x₃. Then set f = x₁ − 2Q(Q(x₂ + x₄) + x₁x₄). Multiplied out, f has 16 terms and degree 5. At (0, 0, 0, 0) it equals 0. At (1, 0, 0, 0) it equals 1. At (1, 1, 1, 1), Q equals 1 and f equals −5.

Add a fifth variable, w, that f does not even use. The paper builds an explicit change of all five variables that turns f into x₁. It then proves that no change of the original four variables does thisSource 1. So f is a stable coordinate [a polynomial that becomes a coordinate once spare variables are added]. Yet it is not a coordinate on its own. A Lean proof is listed for both halves.

The stable coordinate conjecture says this cannot happen. It predicts that a stable coordinate was a coordinate all along. Shpilrain and Yu stated the one-variable form as a conjecture in 2002. They also showed that it holds over the complex numbers in two and three variables. With one variable it is elementary. So four variables is the first place a counterexample could existSource 2. One spare variable is already enough here.

The example also answers a second question. A fiber of f is the set of points where f takes one fixed value. For a coordinate, every fiber is a flat copy of affine three-space [ordinary three-dimensional complex space]. The paper shows every fiber of f is also a copy of three-spaceSource 3. Yet no change of variables of four-space carries any of them to a coordinate hyperplaneSource 3. That is a flat slice such as x₁ = 0. Mathematicians call such a fiber nonrectifiable [impossible to straighten by any polynomial change of the whole space]. The Abhyankar–Sathaye conjecture predicts that this never happens. In two variables the Abhyankar–Moh–Suzuki theorem says the answer is yes. The example says no in four-dimensional spaceSource 4.

A companion paper from 24 September 2026 had already found a different counterexample to Abhyankar–Sathaye. Its polynomial F also has four variables, and the set where F = 0 is a copy of three-spaceSource 5. F fails to be a coordinate for a simple reason. At the point (2, 0, −1/2, 1/2), F equals −1 and all four of its slopes are zeroSource 6. A coordinate can never have such a flat point, because the change of variables would then have no inverse thereSource 7. That paper claims the three-space shape for the zero fiber onlySource 5.

The October polynomial f cannot be caught by that test. It becomes a coordinate once w is added, and the same slope argument then rules out any flat point. That is the real obstacle. Earlier work found polynomials that become coordinates after adding variables. Lewis proved in 2013 that one well-known test polynomial, the second Vénéreau polynomial, is a coordinate. Blanc and Poloni later proved stability for a wider family. Showing that a polynomial is a stable coordinate is explicit work. Showing that it is not a coordinate needs a different kind of argumentSource 8.

Here is the idea in plain terms. If f were a coordinate, a certain sliding symmetry would have to survive on a simpler ring built from f. The paper then shows that no such symmetry exists. One quantity in it would have to be a polynomial and also carry a negative weightSource 9. Both cannot be true. The strategy adapts an earlier OpenAI preprint on a related question, called cancellationSource 10. This example, however, does not itself give a cancellation counterexample. Its surrounding space is ordinary four-space, and the problem sits in the polynomial fSource 11.

Adding more unused variables keeps the Abhyankar–Sathaye failure in every dimension of four or moreSource 12. The three-variable case is not covered. How sure is this? The Lean scope note lists a formal version of the main claims. It covers the degree-five polynomial and its coordinate status with and without w. It also covers the claim that every fiber is three-space and cannot be straightened. It also covers the Abhyankar–Sathaye failure in every dimension of at least four. No specialist review has been published.

What it doesn't show. This does not settle the Abhyankar–Sathaye question in three variables, and it does not give a cancellation counterexample.