Cyclic Rigidity in Dimension Two:
A Newton-Edge Obstruction for a Pseudoreflection Normal Form

1. The remaining two-dimensional problem

For a polynomial map \(F=(P,Q):k^2\to k^2\), write

The two-dimensional Jacobian Conjecture asks whether \(\operatorname{Jac}(P,Q)\in k^\times\) forces \(F\) to be a polynomial automorphism when \(k\) has characteristic zero. After the appearance of a three-dimensional counterexample in July 2026, dimension two remains the unresolved case. The present report does not settle that case. It settles a natural equivariant subproblem that arose from attempting to export the known three-dimensional mechanism to the plane.

Fix \(m\ge2\), extend scalars when necessary to contain a primitive \(m\)-th root of unity \(\zeta\), and define source and target representations

The normal form

satisfies \(F\circ\rho_s(\zeta)=\rho_t(\zeta)\circ F\); it does not generally commute with one identical diagonal action. Equivalently, after the target-coordinate swap \(S(P,Q)=(Q,P)\),

commutes with \(\operatorname{diag}(\zeta,1)\). The swap changes the Jacobian only by a sign. The theorem itself also includes \(m=1\), where no nontrivial cyclic action is present. The case \(m=2\) is the reflection normal form developed in Report II.

2. Context and relation to prior work

Miyanishi proved an equivariant Jacobian theorem over \(\mathbb C\) for small finite subgroups of \(\mathrm{GL}(2,\mathbb C)\) of even order [2]. The pure cyclic group generated by \(\operatorname{diag}(\zeta,1)\) is generated by pseudoreflections and is not small. In Miyanishi’s reduction by the subgroup generated by pseudoreflections, this pure cyclic group has trivial residual quotient; parity of the original order alone therefore does not directly settle the normal form studied here. The present argument is direct, works over every characteristic-zero field, and is uniform in \(m\).

Report II derived, for \(m=2\), a quotient identity and a dichotomy: constant \(g\) gives an automorphism, while a nonconstant \(g\) would force collisions. Exact searches subsequently excluded 7,824 bounded reflection ansätze, and a separate recurrence excluded a single active \(u\)-mode for arbitrary degree. Those results were evidence for rigidity, not a proof. The present theorem supplies the missing global argument.

3. The cyclic Keller equation

Differentiate

Since \(u_x=mx^{m-1}\), direct calculation gives

Thus the Keller condition is the polynomial partial differential equation

At \(u=0\), the equation becomes

Because the product of two polynomials is a nonzero scalar, both factors are scalars. Consequently

This forced affine boundary term is what guarantees that the highest weight of \(H\) used below is at least one.

4. Main theorem

The theorem is unbounded in every degree parameter. No assumption is made about sparsity, homogeneity, irreducibility, the number of nonzero \(u\)-modes of \(g\), or the support of \(H\).

5. Proof by the maximal Newton edge

5.1 The maximal slope of \(g\)

Write

where \(a_0\in k^\times\). Suppose for contradiction that \(K\ge1\). Define

Write \(\rho=p/q\) in lowest terms, with \(p\ge0\) and \(q\ge1\), and assign the rational weights

A monomial \(u^jy^d\) has weight \(d-\rho j\). By maximality of \(\rho\), every nonconstant-in-\(u\) monomial of \(g\) has weight at most zero. The weight-zero terms satisfy \(qd=pj\); coprimality forces \(j=qn\), \(d=pn\). Therefore the weight-zero edge is

for a polynomial \(A\in k[t]\) satisfying

5.2 The highest edge of \(H\)

Let \(\sigma\) be the largest weight appearing in \(H\), and let \(H_{[\sigma]}\) be the corresponding weighted-homogeneous component. Since \(H(0,y)\) has a nonzero linear term,

Choose the unique residue \(0\le r

All exponent pairs on this edge differ by multiples of \((q,p)\). Hence

for some nonzero \(\Phi\in k[t]\). Moreover, \(s>0\), because \(s=0\) would give \(\sigma=-\rho r\le0\).

5.3 Isolation of the highest Keller component

Weighted differentiation changes weight by

Both terms in the cyclic Keller operator have highest possible weight \(\sigma-1\). Any negative-weight component of \(g\) lowers that weight, so the top component depends only on \(H_{[\sigma]}\) and \(g_{[0]}\). Because the right side \(c\) has weight zero:

• if \(\sigma>1\), the component of weight \(\sigma-1\) must vanish;
• if \(\sigma=1\), the component of weight zero must equal \(c\).

5.4 Chain-rule calculation

For \(t=u^qy^p\),

Using \(H_{[\sigma]}=u^ry^s\Phi(t)\), \(g_{[0]}=A(t)\), and \(q\sigma=qs-pr\), direct differentiation yields

where

The apparent complexity of the two-variable PDE has collapsed to a one-variable polynomial identity.

5.5 The terminal coefficient cannot cancel

Put

If \(a_D\) and \(\phi_N\) denote the leading coefficients, then the coefficient of \(t^{N+D}\) in \(B\) is

Every quantity inside the parentheses is nonnegative, and \(s>0\). In characteristic zero,

Therefore \(B\ne0\).

Both cases are impossible. Hence the assumption that \(g\) contains a positive power of \(u\) is false, and

This proves Theorem 1.

6. Classification and explicit inverse

Once \(g=a_0\in k^\times\), the cyclic Keller equation reduces to

Therefore

for a polynomial \(h\in k[u]\), and every map in the class is exactly

For target coordinates \((P,Q)=F(x,y)\),

The theorem is therefore a classification, not merely an injectivity result: all Keller maps in this cyclic pseudoreflection normal form are elementary triangular automorphisms after a coordinate swap and scaling.

7. Quotient identity and collision dichotomy

Define the target invariant

A direct calculation gives the quotient identity

Under \(\det JF=c\),

Raising to the \(m\)-th power and using \(K=ug^m\) yields

For \(m\ge2\), a nonconstant solution would have been immediately dangerous. Since \(g(0,y)=a_0\ne0\), every zero \((u_0,y_0)\) of a nonconstant \(g\) has \(u_0\ne0\). Choosing an \(m\)-th root \(x_0\) of \(u_0\), the distinct points

python -m pip install -r requirements.txt
python run_all.py

full cyclic rigidity: PASS
  determinant identity checks:       28
  terminal coefficient checks:   96,840
  bounded multimode exclusions:     120
independent dictionary checks:     1,952
adversarial Groebner cases:        4,920
single-u-mode degree cases:          210