Results on polynomial maps and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Finite fibres — Proposition 2.2. Keller maps of k^n, k algebraically closed of characteristic zero: every fibre is finite.
• Counting preimages — Theorem 3.3. Keller maps (more generally, polynomial maps with algebraically independent components) over an algebraically closed field of characteristic zero: outside a hypersurface, every point has exactly deg F preimages, where deg F is the degree of the function field extension.
• Injectivity criterion — Theorem 4.1. Keller maps of k^n, k algebraically closed of characteristic zero: a polynomial automorphism exactly when injective, exactly when of degree one. By Remark 4.2 the inverse is defined over any field of definition.
• Wang's theorem — Theorem 5.2. Keller maps with components of degree at most two over a field of characteristic zero are polynomial automorphisms; false in characteristic two.
• Ax-Grothendieck — Theorem 6.1. Injective polynomial maps of k^n to itself, k algebraically closed of any characteristic, are surjective.
• The Jacobian determinant — Theorem 3.1. The explicit polynomial map F of three-space, with rational coefficients: its Jacobian determinant is identically -2.
• Failure of the Jacobian conjecture — Corollary 4.2. Every dimension n >= 3 and every field of characteristic zero: a Keller map that is not injective, hence has no inverse.
• Invariants of a coordinate — Proposition 1.2. Coordinates of a polynomial ring in n variables over any field: every fibre is affine (n-1)-space, and the gradient has no zero over any extension field.
• Lifting lemma — Lemma 2.1. Any commutative ring B with elements h, x, y, s such that x^2 + y^3 = hs and hB + yB = B: the B-algebra cut out by the perturbed cusp parametrization is a polynomial ring in one variable over B.
• An affine-space hypersurface — Theorem 3.1. Characteristic zero: the explicit polynomial F of degree 17 in four variables has quotient ring isomorphic to a polynomial ring in three variables, by explicit mutually inverse maps.
• Failure of the Abhyankar-Sathaye conjecture — Corollary 4.2. Every n >= 4, characteristic zero: a polynomial in n variables whose zero set is affine (n-1)-space but which is not a coordinate; the fibre through its critical point is not affine space.
• Degree three — Theorem 5.4. The counterexample has degree three over an algebraically closed field of characteristic zero; its fibres in the open set z1(1+z1z2) != 0 correspond to roots of the cubic 2a l^3 - b l^2 + 2l - c.
• Derivations and additive actions — Proposition 2.3. Characteristic zero: actions of the additive group on Spec A are exactly the exponentials of locally nilpotent derivations of A.
• Order and factorial closure — Proposition 3.2. Locally nilpotent derivations of a domain of characteristic zero: the order is additive, the kernel is factorially closed and contains the units.
• Induced homogeneous derivation — Proposition 6.2. A domain with a degree function and exact generators: a nonzero locally nilpotent derivation induces a nonzero homogeneous one on the associated graded ring, killing initial forms of invariants.
• Picard group of a polynomial ring — Proposition 2.1. Regular Noetherian domains C: Pic(C) -> Pic(C[t]) is an isomorphism.
• Lifting additive actions — Theorem 3.2. C a regular Noetherian domain containing a field of characteristic zero, T a Laurent bundle algebra over C: every locally nilpotent derivation of C extends uniquely to a weight-preserving one of T.
• An affine cylinder — Theorem 4.1. Characteristic zero: the explicit hypersurface A = P/(H) in affine five-space satisfies A[w] = k^[5], with explicit coordinates.
• Failure of cancellation — Theorem 9.1. Every field of characteristic zero: A[w] is a polynomial ring in five variables but A is not a polynomial ring in four.
• Mason-Stothers — Theorem 7.1. Pairwise coprime polynomials X + Y = Z over a field of characteristic zero, not all constant: the maximal degree is less than the number of distinct roots of XYZ.
• A transfer lemma — Lemma 1.1. B noetherian, d = e d' in B, S in B[lam]: lam + d(S(lam) + e w) is a coordinate of B[lam, w], with the explicit partner w - d'(S(lam) + e w) deltaS(lam, F) built from the divided difference of S.
• A stable coordinate in five variables — Theorem 3.1. Characteristic zero: the polynomial H of the cancellation counterexample is a coordinate of k[p,s,u,F,J,w] but not of k[p,s,u,F,J].
• A stable coordinate that is not a coordinate — Theorem 7.1. Characteristic zero: f = x1 - 2Q(Q(x2+x4) + x1x4), Q = x2^2 - x4^2 + x1x3, is a coordinate after adjoining one variable but not before; every fibre is affine three-space and none is rectifiable.
• Smoothness of a hypersurface — Lemma 2.1. If the partial derivatives of f generate the unit ideal of R[X]/(f), then R[X]/(f) is a smooth R-algebra.
• Flatness of a hypersurface — Lemma 2.2. Over a noetherian ring R, R[X]/(f) is flat when the image of f in kappa(p)[X] is nonzero for every prime p.
• A residual coordinate — Proposition 3.1. The polynomial H of the cancellation counterexample is a coordinate of kappa(p)[s,u,F,J] for every prime p of k[p].
• Two smooth A3-fibrations — Theorem 4.1. k[p,s,u,F,J] over k[p,q] through q -> H, and k[p,s,u,F,J]/(H) over k[p], are flat and smooth, and all their fibre rings are polynomial rings in three variables.
• Nontrivial over a discrete valuation ring — Theorem 5.1. k[p,s,u,F,J]/(H) localized at p = 0 is not a polynomial ring over k[p]_(p), and k[p,s,u,F,J] is not a polynomial ring over k[p,q]_(p,q).
• Affine-space fibrations that are not locally trivial — Corollary 5.2. Neither fibration is Zariski-locally trivial or the symmetric algebra of a projective module; so an A^3-fibration over a regular ring, even over a discrete valuation ring containing Q, need not be a polynomial ring.

Reading order

Read the lessons in order. The first proves that a Keller map is an automorphism exactly when it is injective, by counting preimages through the degree of the function field extension. The second proves, by a computation in coordinates adapted to the map, that an explicit map of three-space is a Keller map of degree three that is not injective. The third shows that a hypersurface isomorphic to affine space need not be a coordinate hyperplane. Lessons four and five develop locally nilpotent derivations and their lifts to line bundles; lessons six to nine use them to show that cancellation fails in dimension four and that a stable coordinate need not be a coordinate, in five and in four variables. The last lesson shows that the same polynomial gives smooth affine-space fibrations that are not locally trivial.