Differential conditions to verify the Jacobian Conjecture

Ludwik M. Drużkowski; Halszka K. Tutaj

Annales Polonici Mathematici (1992)

  • Volume: 57, Issue: 3, page 253-263
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Ludwik M. Drużkowski, and Halszka K. Tutaj. "Differential conditions to verify the Jacobian Conjecture." Annales Polonici Mathematici 57.3 (1992): 253-263. <http://eudml.org/doc/275935>.

@article{LudwikM1992,
abstract = {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.},
author = {Ludwik M. Drużkowski, Halszka K. Tutaj},
journal = {Annales Polonici Mathematici},
keywords = {jacobian conditions; global injectivity; global stability; Jacobian conjecture; polynomial mapping; autonomous system of differential equations; globally asymptotically stable},
language = {eng},
number = {3},
pages = {253-263},
title = {Differential conditions to verify the Jacobian Conjecture},
url = {http://eudml.org/doc/275935},
volume = {57},
year = {1992},
}

TY - JOUR
AU - Ludwik M. Drużkowski
AU - Halszka K. Tutaj
TI - Differential conditions to verify the Jacobian Conjecture
JO - Annales Polonici Mathematici
PY - 1992
VL - 57
IS - 3
SP - 253
EP - 263
AB - 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.
LA - eng
KW - jacobian conditions; global injectivity; global stability; Jacobian conjecture; polynomial mapping; autonomous system of differential equations; globally asymptotically stable
UR - http://eudml.org/doc/275935
ER -

References

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  1. [B] N. E. Barabanov, On Kalman's problem, Sibirsk. Mat. Zh. 29 (3) (1988), 2-11 (in Russian). 
  2. [BR] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203. Zbl0107.14602
  3. [BCR] J. Bochnak, M. Coste et M.-F. Roy, Géométrie Algébrique Réelle, Springer, Berlin 1987. 
  4. [D] F. Dillen, Polynomials with constant Hessian determinant, J. Pure Appl. Algebra 71 (1991), 13-18. Zbl0741.12001
  5. [E] A. van den Essen, A note on Meisters and Olech's proof of the global asymptotic stability Jacobian conjecture, Pacific J. Math. 151 (1991), 351-356. Zbl0752.12002
  6. [H] P. Hartman, Ordinary Differential Equations, Wiley, New York 1964. Zbl0125.32102
  7. [HO] P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc. 104 (1962), 154-178. 
  8. [KR] K. Kurdyka and K. Rusek, Surjectivity of certain injective semialgebraic transformations of ℝⁿ, Math. Z. 200 (1988), 141-148. Zbl0641.14010
  9. [Ł] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel 1991. Zbl0747.32001
  10. [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305-317. Zbl0096.28802
  11. [MO] G. H. Meisters and C. Olech, Solution of the global asymptotic stability Jacobian conjecture for the polynomial case, in: Analyse Mathématique et Applications, Gauthier-Villars, Paris 1988, 373-381. Zbl0668.34048
  12. [MO1] G. H. Meisters and C. Olech, A Jacobian condition for injectivity of differentiable plane maps, Ann. Polon. Math. 51 (1990), 249-254. Zbl0734.26008
  13. [Md] D. Mumford, Algebraic Geometry, I. Complex Projective Varieties, Springer, Berlin 1976. Zbl0356.14002
  14. [O] C. Olech, On the global stability of an autonomous system on the plane, Contributions Differential Equations 1 (1963), 389-400. 
  15. [P] T. Parthasarathy, On Global Univalence, Lecture Notes in Math. 977, Springer, Berlin 1983. 

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