On Jacobian \(n\)-tuples in characteristic \(p\)
Let \(k\) be a field of characteristic \(p>0\), \(A = k[X_ 1, \dots, X_ n]\), and \(L=k (X_ 1, \dots, X_ n)\). The determinant of the \(n \times n\) Jacobian matrix of \(F_ 1, \dots, F_ n \in A\) with respect to \(X_ i\) is denoted by \(j(F_ 1, \dots, F_ n)\). Then \((F_ 1, \dots, F_ n)\) is a Jacobian \(n\)-tuple if \(j(F_ 1, \dots, F_ n)\) is in \hbox{\(k^* = k- \{0\}\)}. The authors give a new characterization of Jacobian \(n\)-tuples in terms of the differential operator \(\nabla = (\partial/ \partial_ 1)^{p-1} \cdots (\partial/ \partial_ n)^{p- 1}\) in the hope of shedding some light on the famous Jacobian conjecture (in characteristic zero). The main result is: Theorem. The following conditions are equivalent: (1) \((F_ 1, \dots, F_ n)\) is a Jacobian \(n\)-tuple. (2) \(A=k [X_ 1^ p, \dots, X_ n^ p, F_ 1, \dots, F_ n]\). (3) \(\{F_ 1^{k_ 1}, \dots, F_ n^{k_ n}| 0 \leq k_ i \leq p-1\}\) is a free basis of the \(k[X_ 1^ p, \dots, X_ n^ p]\)-module \(A\). (4) For any \(h \in L\), we have \(h=c \sum_{0 \leq r \leq p-1} F_ i^ r d_ i^{p-1} (F_ i^{p-r-1} h)\) for some \(c \in k^*\). (5) \(\nabla = cd_ 1^{p-1} \cdots d_ n^{p-1}\) for some \(c \in k^*\). (6) \(\nabla (F_ 1^{k_ 2} \cdots F_ n^{k_ n}) = 0\) if \(0 \leq k_ i<p -1\) for some \(i\), and \(\nabla (F_ 1^{p-1} \cdots F_ n^{p-1})\) is in \(k^*\). Here \(d_ i(h) = j(F_ 1, \dots, F_{i-1}, h, F_{i+1}, \dots, F_ n)\). (The equivalence of (1), (2), (3) is due to \textit{Nousainen}).
scientific article; zbMATH DE number 218644