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Let / : R -> TBL" be a C -vector field with /(0) = 0 . For p e 1" let Jf(p) denote its Jacobian matrix evaluated at p. Then it is a well-known result, due to Lyapunov, that the origin is a locally asymptotic rest point of the non-linear autonomous system of ordinary differential equations x = f(x) if the origin is a locally asymptotic rest point of the linearized system y = Jf(0)y (or equivalently if all eigenvalues of the matrix Jf(0) have negative real parts). In 1960 it was conjectured by Markus and Yamabe that the origin is a globally asymptotic rest point x = f(x) if for each p e R the orgin is a locally asymptotic rest point of the linearized system y — Jf(p)y. Until now this conjecture is still open. However in 1988 Meisters and Olech proved this conjecture for two-dimensional polynomial vector fields / : M —• M. The proof is an immediate consequence of earlier results of Olech, (1963) and the proposition below. The main result of this paper (Theorem 1) generalizes the proposition to polynomial maps F: k —> k having the property that det JF(x) φ 0 for all x € k (k is a field of characteristic zero).

A note on Meisters and Olech’s proof of the global asymptotic stability Jacobian conjecture