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The Jacobian conjecture and affine spaces

Polynomial maps and affine spaces: Keller maps and the reduction of the Jacobian conjecture to injectivity, with a complete hand proof that an explicit map of three-space with Jacobian determinant -2 identifies three points; coordinates and a degree-17 polynomial in four variables whose zero set is affine three-space but which is not a coordinate; locally nilpotent derivations, additive actions and their lifts to line bundles; a fourfold whose cylinder is affine five-space but which is not affine four-space, so that Zariski cancellation fails in dimension four in characteristic zero; a polynomial in five variables that becomes a coordinate after adjoining one variable, with explicit complementary coordinates, but is not a coordinate; a degree-five stable coordinate in four variables that is not a coordinate; and smooth affine-space fibrations over the line, the plane and a discrete valuation ring, all of whose fibres are affine three-space, which are not locally trivial.

Ten lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Keller maps and the Jacobian conjecture
  2. A counterexample in dimension three
  3. Coordinates and the Abhyankar–Sathaye problem
  4. Locally nilpotent derivations
  5. Additive actions on line bundles
  6. A fourfold whose cylinder is affine space
  7. Cancellation fails in dimension four
  8. A stable coordinate in five variables
  9. A stable coordinate that is not a coordinate
  10. Affine-space fibrations that are not locally trivial

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites