Let f : ℝn→ ℝn be a
C1-vector field with f(0) = 0. For p ∈ ℝn 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 ẋ= f(x) if the origin is a locally asymptotic rest point of the
linearized system ẏ= 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 ẋ= f(x) if for each p ∈ ℝn the orgin is a locally asymptotic
rest point of the linearized system ẏ= 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 f : ℝ2→ ℝ2. 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 : kn→ kn
having the property that detJF(x)≠0 for all x ∈ kn (k is a field of characteristic
zero).
