On locally invertible systems of polynomials (Q1335055)

Let \(K\) be a commutative integral domain, \(K^*\) its group of units and \(F = (F_ 1, \dots, F_ n) \in K[X]^ n\), \(X = (X_ 1, \dots, X_ n)\). \(F\) is called locally invertible if for every point \(a \in K^ n\) the system of polynomials \(F_ a (X) : = F(X + a) - F(a)\) has a formal inverse, i.e. there exists \(G_ a \in K [[X]]^ n\) such that \(F_ a \circ G_ a = G_ a \circ F_ a = X\) (this is equivalent to the jacobian condition, i.e. \(\text{Jac} (F) (a) \in K^*\) for every \(a \in K^ n)\). The main theorem of the paper is the following upper bound for the cardinality of the fibers of \(F\): if \(K\) is of characteristic zero and \(F \in K [X]^ n\) is locally invertible, then the field \(K(X)\) is a finite algebraic extension of the field \(K(F)\) and for any point \(b \in K^ n\) holds: \(\# F^{-1} (b) \leq [K(X) : K(F(X))]\).