An algebraic characterization of entire polynomial diffeomorphisms

Abstract

AbstractIt is shown that a polynomial map $$F: {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ F : R n → R n with nowhere zero Jacobian determinant is invertible if and only if an explicit auxiliary polynomial system admits only the trivial solution. The main corollary is a concrete invertibility criterion in the Jacobian conjecture. The proof, conceptually related to differential geometry, represents a simple but infrequent application of differential equations to algebra.