Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms

Author: Arno van den Essen
Journal: Proc. Amer. Math. Soc. 116 (1992), 861-871
MSC: Primary 13B10; Secondary 14E09, 34A99, 34C99
MathSciNet review: 1111440
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a very simple proof of the fact that the Lorenz equations and the Maxwell-Bloch equations do not have a polynomial flow. We also give an algorithm to decide if a two-dimensional vector field over $ \mathbb{R}$ has a polynomial flow and how to compute the solutions (in case the vector field has a polynomial flow).

References [Enhancements On Off] (What's this?)

  • [1] H. Bass, A non-triangular action of $ {G_a}$ on $ {\mathbb{A}^3}$, J. Pure Appl. Algebra 33 (1984), 1-5. MR 750225 (85j:14086)
  • [2] H. Bass and G. H. Meisters, Polynomial flows in the plane, Advances in Math. 55 (1985), 173-208. MR 772614 (86c:58127)
  • [3] B. Coomes, Polynomials flows, symmetry groups, and conditions sufficient for injectivity of maps, PhD thesis, Univ. of Nebraska-Lincoln, 1988.
  • [4] -, The Lorenz system does not have a polynomial flow, J. Differential Equations 82 (1989), 386-407. MR 1027976 (91b:58213)
  • [5] B. Coomes and V. Zurkowski, Linearization of polynomial flows and spectra of derivations, J. Dynamics Differential Equations 3 (1990), 29-66. MR 1094723 (92m:34002)
  • [6] A. van den Essen, $ \mathcal{D}$-modules and the Jacobian Conjecture, Report 9108, Catholic Univ. Nijmegen; Proc. Internat. Conf. $ \mathcal{D}$-Modules and Microlocal Geometry, Lisbon, Portugal, October 1990 (to appear).
  • [7] H. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161-174. MR 0008915 (5:74f)
  • [8] W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde (3) I (1953), 33-41. MR 0054574 (14:941f)
  • [9] G. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), 679-705. MR 683862 (84c:58048)
  • [10] -, Polynomial flows on $ {\mathbb{R}^n}$, Banach center publications, Vol. 23, Warzawa, 1989.
  • [11] P. Nousiainen and M. Sweedler, Automorphisms of polynomial and power series rings, J. Pure Appl. Algebra 29 (1983), 93-97. MR 704289 (84i:13004)
  • [12] R. Rentschler, Opérations du groupe additif sur le plane affine, C. R. Acad. Sci. Paris 267 (1968), 384-387. MR 0232770 (38:1093)
  • [13] M. Smith, Stably tame automorphisms, J. Pure Appl. Algebra 58 (1989), 209-212. MR 1001475 (90f:13005)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13B10, 14E09, 34A99, 34C99

Retrieve articles in all journals with MSC: 13B10, 14E09, 34A99, 34C99

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society