Study on affine algebraic varieties and extension of the base field

The Jacobian conjecture asserts that a polynomial map of the ndimensional affine space over the complex number field defined by n polynomials in n variables with the Jacobian nonzero constant must be an isomorphism. The conjecture is still open for any n>1. We study the conjecture from the algebraic and topological point of view and give several conditions equivalent to the conjecture.