Summary: A polynomial map \(F:\mathbb{R}^2\to\mathbb{R}^2\) is said to satisfy the Jacobian condition if \(\forall (X,Y)\in \mathbb{R}^2\), \(J(F)(X,Y)\neq 0\). The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map \(F:\mathbb{R}^2\to \mathbb{R}^2\) that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only \(X\)- or \(Y\)-finite asymptotic values. We prove that a \(Y\)-finite asymptotic value can be realized by \(F\) along a rational curve of the type \((X^{-k},A_0+A_1X+\cdots+ A_{N-1}X^{N-1}+ YX^N)\), where \(X\to 0\), \(Y\) is fixed and \(K,N>0\) are integers. More precisely we prove that the coordinate polynomials \(P(U,V)\) of \(F(U,V)\) satisfy finitely many asymptotic identities, namely, identities of the following type, \(P(X^{-k},A_0+ A_1X+\cdots+ A_{N-1} X^{N-1}+ YX^N)= A(X,Y)\in R[X,Y]\), which ‘capture’ the whole set of asymptotic values of \(F\).