Power-Similarity: Summary of First Results

Around 1960, Markus & Yamabe [1] conjectured that, for each n ≥ 2, every rest point x0 of a nonlinear, class C1, n-dimensional system of differential equations ẋ = F (x) is globally asymptotically stable if the eigenvalues of the jacobian matrix F (x) have negative real parts at every point x in R. At that time, Olech [2] connected this problem with the question of injectivity of the mapping x 7→ F (x). An older question, evidently first raised by Keller [3] in 1939 for polynomials over the integers but now also raised for complex polynomials, and, as such, known as “The Jacobian Conjecture” (JC), asks whether polynomial maps F (x) with nonzero constant jacobian determinant det F (x) need be injective (and also surjective with polynomial inverse). It suffices to prove injectivity because in 1960–62 it was proved, first in dimension 2 by Newman [4] and then in all dimensions by Bia lynicki-Birula and Rosenlicht [5], that, for polynomial maps, surjectivity follows from injectivity; and furthermore, under the hypothesis det F (x) = const 6= 0, the inverse F(x) will be polynomial, at least in the complex case, if the polynomial map is bijective. About 1980 it was shown by Bass, Connell, and Wright [6], and also by Yagzhev [7], that to prove JC in every dimension n ≥ 2 it suffices to prove it (but also in every dimension n ≥ 2) for the special case of maps of the form F (x) = x−H(x), where H(x) is homogeneous of degree three. Using this “Reduction of Degree” result, it is easy to show that the truth of the Markus-Yamabe Conjecture implies the truth of JC: For suppose that, in some dimension n, there is a mapping F (x) :≡ x − H(x) that is not injective, where H(tx) ≡ tH(x) and H (x) ≡ 0. (This nilpotence of H (x) is equivalent to Keller’s condition that F (x) have a nonzero constant jacobian determinant. For a proof, see e.g., [8, Lemma 1(c)].) Then F (x1) = F (x2) for some x1 6= x2 in C. So, by the Markus-Yamabe Conjecture, both x1 and x2 are globally asymptotic rest points of the system