Research directions

Research directions#

The counterexample closes the all-dimensional conjecture but opens a more concrete research program.

Recover the search family#

The map is affine-linear in \(z\):

\[ F(x,y,z)=A(x,y)z+B(x,y). \]

Its \(z\)-coefficient row

\[ ((1+xy)^3,\,3x(1+xy)^2,\,-x^3) \]

is built from the unimodular pair \(x,1+xy\). Understanding the parameter space of such rows and solving the constant-Jacobian equations coefficient by coefficient may yield smaller examples or a classification.

Minimize complexity#

Natural optimization targets are:

  • maximum coordinate degree;

  • total monomial count;

  • coefficient height;

  • a homogeneous or sparse normal form;

  • the dimension introduced by cubic stabilization.

The present normalized example has coordinate degrees \((4,6,7)\) and integral coefficients. Classical results show that maximum degree \(2\) is impossible, but the gap between \(2\) and \(7\) in dimension three is now concrete.

Make stabilization explicit#

Apply Bass–Connell–Wright/Yagzhev and Drużkowski reductions algorithmically to this exact map. Record the resulting dimensions, coefficient growth, and the transported collision. This would turn an existential corollary into a compilable cubic counterexample.

Plane obstruction#

The missing step is now precise: affine-linearity, affine two-planes of marked forms, polynomial graph restrictions, linear target slices, and the natural equivariant nonlinear target shears are all ruled out. The active construction target is a non-Galois degree-six cusp cover whose finite normalization is not affine two-space but contains affine two-space as a proper étale open. See the exact obstructions, near-models, and the focused program.

Geometry and topology#

  • Compute the topology and stratification of the discriminant complement.

  • Describe the \(S_3\) monodromy of the three inverse branches.

  • Study the embedding of the marked-root space into compactifications and the divisors responsible for escape.

  • Write the induced rank-three Weyl algebra endomorphism explicitly.

Formal verification#

Translate the structural proof into Lean, Isabelle, or Coq. The core requires only polynomial normalization, the chain rule, and a three-point inequality, making it unusually suitable for a short proof-assistant certificate.