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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  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
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