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Differential conditions to verify the Jacobian Conjecture

Ludwik M. Drużkowski, Halszka K. Tutaj (1992)

Annales Polonici Mathematici

Let F be a polynomial mapping of ℝ², F(O) = 0. In 1987 Meisters and Olech proved that the solution y(·) = 0 of the autonomous system of differential equations ẏ = F(y) is globally asymptotically stable provided that the jacobian of F is everywhere positive and the trace of the matrix of the differential of F is everywhere negative. In particular, the mapping F is then injective. We give an n-dimensional generalization of this result.

Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

Kostov, Vladimir (2002)

Serdica Mathematical Journal

∗ Research partially supported by INTAS grant 97-1644A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study...

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