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)

 
 

 

On the torsion part of $ {\bf C}\sp {[n]}$ with respect to the action of a derivation


Author: Brian Coomes
Journal: Proc. Amer. Math. Soc. 123 (1995), 2191-2197
MSC: Primary 34A20; Secondary 12H05, 34C99
DOI: https://doi.org/10.1090/S0002-9939-1995-1273485-7
MathSciNet review: 1273485
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the torsion part of $ {\mathbb{C}^{[n]}}$ with respect to the action of a derivation is algebraically closed in $ {\mathbb{C}^{[n]}}$ if the flow associated with the derivation is analytic on $ \mathbb{C} \times {\mathbb{C}^n}$. We also present a connection between this result and Keller's Jacobian conjecture.


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

  • [1] H. Bass, Differential structure of étale extensions of polynomial algebras, Commutative Algebra (Proceedings of a microprogram held June 15-July 2, 1987), Math. Sci. Res. Inst. Publ., vol. 15, Springer-Verlag, New York, 1989, pp. 69-109. MR 1015514 (90m:12009)
  • [2] H. Bass, E. H. Connell, and D. Wright, The Jacobian conjecture, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 287-330. MR 663785 (83k:14028)
  • [3] H. Bass and G. H. Meisters, Polynomial flows in the plane, Adv. Math. 55 (1985), 173-203. MR 772614 (86c:58127)
  • [4] E. Connell and J. Drost, Conservative and divergence free algebraic vector fields, Proc. Amer. Math. Soc. 87 (1983), 607-612. MR 687626 (84i:13014)
  • [5] B. Coomes, The Lorenz system does not have a polynomial flow, J. Differential Equations 82 (1989), 386-407. MR 1027976 (91b:58213)
  • [6] -, Polynomial flows on $ {\mathbb{C}^n}$, Trans. Amer. Math. Soc. 320 (1990), 493-506. MR 998353 (90k:58180)
  • [7] -, P-symmetries of two-dimensional p-f vector fields, Differential Integral Equations 5 (1992), 461-480. MR 1148229 (93b:34012)
  • [8] B. Coomes and V. Zurkowski, Linearization of polynomial flows and spectra of derivations, J. Dynamics Differential Equations 3 (1991), 29-66. MR 1094723 (92m:34002)
  • [9] O. Keller, Ganze cremona-transformationen, Monatsh. Math. Phys. 47 (1939), 299-306. MR 1550818
  • [10] G. H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), 679-705. MR 683862 (84c:58048)
  • [11] -, Polynomial flows on $ {\mathbb{R}^n}$, Dynamical Systems and Ergodic Theory, Papers from the 28th semester at the Stefan Banach International Mathematical Center (September 16-December 18, 1986), Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 9-24. MR 1102697 (92b:58060)
  • [12] G. H. Meisters and C. Olech, Global asymptotic stability for plane polynomial flows, Časopis Pěst. Mat. 111 (1986), 123-126. MR 847311 (87k:34085)
  • [13] -, A poly-flow formulation of the Jacobian conjecture, Bull. Polish Acad. Sci. Math. 35 (1987), 725-731. MR 961711 (89j:13005)
  • [14] A. van den Essen, Locally nilpotent and locally finite derivations with applications to polynomial flows and polynomial morphisms, Proc. Amer. Math. Soc. 116 (1992), 861-871. MR 1111440 (93a:13003)
  • [15] -, Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms, and $ {\mathcal{G}_a}$-actions, II, Proc. Amer. Math. Soc. 121 (1994), 667-678. MR 1185282 (94i:13005)
  • [16] G. Zampieri, Finding domains of invertibility for smooth functions by means of attraction basins, J. Differential Equations 104 (1993), 11-19. MR 1224119 (94e:26022)
  • [17] V. Zurkowski, Polynomial flows in the plane: a classification based on spectra of derivations, preprint. MR 1339667 (96m:34004)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34A20, 12H05, 34C99

Retrieve articles in all journals with MSC: 34A20, 12H05, 34C99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1273485-7
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society