ON COUNTEREXAMPLES TO KELLER'S PROBLEM

摘要

Let F: K~n → K~n be a polynomial map where K = R or C. Let us denote by J(F) the determinant of the Jacobian of F. The Jacobian conjecture is the following statement: If J(F) never vanishes then the map F is injective. Originally, the conjecture was stated for K = C with polynomials over Z by O. Keller. When K = C the assumption can be rewritten as J(F) ∈ C~* (by the Fundamental Theorem of Algebra) and the conclusion can be rewritten as: F is invertible in the ring C[X_1,..., X_n]. For K = R, injectivity of F implies its surjectivity (a result that was generalized by A. Borel to real algebraic varieties).