# 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

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topLudwik 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

top- [B] N. E. Barabanov, On Kalman's problem, Sibirsk. Mat. Zh. 29 (3) (1988), 2-11 (in Russian).
- [BR] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203. Zbl0107.14602
- [BCR] J. Bochnak, M. Coste et M.-F. Roy, Géométrie Algébrique Réelle, Springer, Berlin 1987.
- [D] F. Dillen, Polynomials with constant Hessian determinant, J. Pure Appl. Algebra 71 (1991), 13-18. Zbl0741.12001
- [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
- [H] P. Hartman, Ordinary Differential Equations, Wiley, New York 1964. Zbl0125.32102
- [HO] P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc. 104 (1962), 154-178.
- [KR] K. Kurdyka and K. Rusek, Surjectivity of certain injective semialgebraic transformations of ℝⁿ, Math. Z. 200 (1988), 141-148. Zbl0641.14010
- [Ł] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel 1991. Zbl0747.32001
- [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305-317. Zbl0096.28802
- [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
- [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
- [Md] D. Mumford, Algebraic Geometry, I. Complex Projective Varieties, Springer, Berlin 1976. Zbl0356.14002
- [O] C. Olech, On the global stability of an autonomous system on the plane, Contributions Differential Equations 1 (1963), 389-400.
- [P] T. Parthasarathy, On Global Univalence, Lecture Notes in Math. 977, Springer, Berlin 1983.

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