The sixtieth anniversary of the Jacobian conjecture: A new approach (Q2724120)

Let \(X=(X_1,\dots, X_n)\), \(F=(F_1,\dots, F_n)\in \mathbb{C}[X]^n\) and \(JF=({\partial F_i\over\partial X_j})\) the Jacobian matrix. Denote by \(JC(\mathbb{C}, n)\) the \(n\)-dimensional Jacobian conjecture: If \(F\in\mathbb{C}[X]^n\) with \(\det(JF) \in\mathbb{C}^*\) then \(\mathbb{C}[F]= \mathbb{C}[X]\).NEWLINENEWLINENEWLINEAn approach of \textit{H. Bass} [in: Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II; Geometry, Prog. Math. 36, 65-75 (1983; Zbl 0527.13004)] to study the Jacobian conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary \(\mathbb{Q}\)-algebra is investigated: \(JC(\mathbb{C},n)\) is equivalent to the following statement:NEWLINENEWLINENEWLINEFor every \(d\geq 1\) there exists a positive integer \(C(n,d)\) such that for any \(\mathbb{Q}\)-algebra \(R\) and any \(F\in\Aut_R R[X]\) with \(\deg(F)\leq d\) and \(\det(JF)=1\) we have \(\deg(F^{-1}) \leq C(n,d)\).NEWLINENEWLINENEWLINEIt is proved that \(JC (\mathbb{C},n)\) is equivalent to the following statement:NEWLINENEWLINENEWLINEFor every \(d\geq 1\) there exists a positive integer \(\overline C(n,d)\) such that for any \(\mathbb{C}\)-algebra \(\mathbb{C}_m:=\mathbb{C}[T]/\langle T^m\rangle\), \(m\geq 2\), and every \(F\in \Aut_{\mathbb{C}_m} \mathbb{C}_m[X]\) with \(\deg(F)\leq d\) and \(\det(JF)=1\) we have \(\deg (F^{-1})\leq \overline C(n,d)\).NEWLINENEWLINENEWLINE\(JC(\mathbb{C},n)\) is equivalent to the following statement: For every \(d\geq 1\) there exists a positive integer \(C_*(n,d)\) such that for every \(m\geq 1\) and every \(D\in\text{Der}_{\mathbb{C}_m} \mathbb{C}_m[X]\) with \(\sum{\partial \over\partial X_i} (DX_i)=0\) and \(D\bmod T=0\) we have: If \(\deg (\exp(D))\leq d\), then \(\deg (\exp(-D)) \leq C_*(n,d)\).