Let \(\phi: \AA^n \to \AA^n\) be a morphism given by \(n\) polynomials \(f_1,\ldots,f_n\) in the variables \(x_1,\ldots,x_n\), and let \(J = \det\qty{ \frac{\partial f_i}{\partial x_j} }\) be the Jacobian polynomial of \(\phi\).
If \(\phi\) is an isomorphism, in which case \(\phi\) is called an automorphism of \(\AA^n\), show that \(J\) is a nonzero constant polynomial.
The converse is an unsolved problem, even for \(n = 2\).
(Characteristic and current status of the converse)
The open Jacobian conjecture is a characteristic-zero problem. In characteristic \(p>0\), the converse already fails for \(n=1\): the polynomial map \[ F(x)=x-x^p \] has derivative \(F'(x)=1\), but \(F(0)=F(1)=0\), so it is not an automorphism.
Hartshorne’s sentence in part (b) records the status when the book was written. As of 2026-09-17, the characteristic-zero two-variable case remains open. The unrestricted higher-dimensional conjecture is no longer open: a counterexample in dimension three was announced in July 2026, and counterexamples in every dimension greater than two are described in Shuhong Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222.
Solution
1. If \(\phi:\AA^n\to\AA^n\) is a polynomial automorphism, then its Jacobian determinant is a unit of \(k[x_1,\ldots,x_n]\).
Let \[ \psi:\AA^n\longrightarrow\AA^n \] be the inverse polynomial morphism, with coordinate functions \(g_1,\ldots,g_n\). Write \(D\phi\) and \(D\psi\) for the two Jacobian matrices. Differentiating \[ \psi\circ\phi=\operatorname{id}_{\AA^n} \] and applying the polynomial chain rule gives \[ D\psi(\phi(x))\,D\phi(x)=I_n. \] Taking determinants yields the polynomial identity \[ J_\psi(\phi(x))\,J_\phi(x)=1. \] Thus \(J_\phi\) has a multiplicative inverse in \(k[x_1,\ldots,x_n]\).
2. Therefore \(J\) is a nonzero constant, proving (a).
The only units of a polynomial ring over a field are the nonzero constants: if nonzero polynomials \(a,b\) satisfy \(ab=1\), total degree gives \[ 0=\deg(ab)=\deg a+\deg b, \] so both degrees are zero. Applying this to step 1 gives \[ J=J_\phi\in k^\times. \]
3. Q.E.D.
Steps <1>1–<1>2 prove the mathematical assertion in part (a). Part (b) is a status statement rather than an additional proof obligation; the qualification below records its present scope.