On the torsion part of $\textbf {C}^ {[n]}$ with respect to the action of a derivation
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- by Brian Coomes
- Proc. Amer. Math. Soc. 123 (1995), 2191-2197
- DOI: https://doi.org/10.1090/S0002-9939-1995-1273485-7
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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
- Hyman Bass, Differential structure of étale extensions of polynomial algebras, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 69–109. MR 1015514, DOI 10.1007/978-1-4612-3660-3_{5}
- Hyman Bass, Edwin H. Connell, and David Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330. MR 663785, DOI 10.1090/S0273-0979-1982-15032-7
- Hyman Bass and Gary Meisters, Polynomial flows in the plane, Adv. in Math. 55 (1985), no. 2, 173–208. MR 772614, DOI 10.1016/0001-8708(85)90020-9
- E. Connell and J. Drost, Conservative and divergence free algebraic vector fields, Proc. Amer. Math. Soc. 87 (1983), no. 4, 607–612. MR 687626, DOI 10.1090/S0002-9939-1983-0687626-5
- Brian A. Coomes, The Lorenz system does not have a polynomial flow, J. Differential Equations 82 (1989), no. 2, 386–407. MR 1027976, DOI 10.1016/0022-0396(89)90140-X
- Brian A. Coomes, Polynomial flows on $\textbf {C}^n$, Trans. Amer. Math. Soc. 320 (1990), no. 2, 493–506. MR 998353, DOI 10.1090/S0002-9947-1990-0998353-7
- Brian Coomes, p-symmetries of two-dimensional p-f vector fields, Differential Integral Equations 5 (1992), no. 2, 461–480. MR 1148229
- Brian Coomes and Victor Zurkowski, Linearization of polynomial flows and spectra of derivations, J. Dynam. Differential Equations 3 (1991), no. 1, 29–66. MR 1094723, DOI 10.1007/BF01049488
- Ott-Heinrich Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), no. 1, 299–306 (German). MR 1550818, DOI 10.1007/BF01695502
- Gary H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), no. 4, 679–705. MR 683862, DOI 10.1216/RMJ-1982-12-4-679
- Karol Krzyżewski (ed.), Dynamical systems and ergodic theory, Banach Center Publications, vol. 23, PWN—Polish Scientific Publishers, Warsaw, 1989. Papers from the Twenty-eighth Semester held in Warsaw, September 16–December 18, 1986. MR 1102697
- Gary H. Meisters and Czesław Olech, Global asymptotic stability for plane polynomial flows, Časopis Pěst. Mat. 111 (1986), no. 2, 123–126 (English, with Russian and Czech summaries). MR 847311, DOI 10.21136/CPM.1986.118270
- Gary H. Meisters and Czesław Olech, A poly-flow formulation of the Jacobian conjecture, Bull. Polish Acad. Sci. Math. 35 (1987), no. 11-12, 725–731 (English, with Russian summary). MR 961711
- Arno van den Essen, Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms, Proc. Amer. Math. Soc. 116 (1992), no. 3, 861–871. MR 1111440, DOI 10.1090/S0002-9939-1992-1111440-5
- Arno van den Essen, Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms and $\scr G_a$-actions. II, Proc. Amer. Math. Soc. 121 (1994), no. 3, 667–678. MR 1185282, DOI 10.1090/S0002-9939-1994-1185282-0
- Gaetano Zampieri, Finding domains of invertibility for smooth functions by means of attraction basins, J. Differential Equations 104 (1993), no. 1, 11–19. MR 1224119, DOI 10.1006/jdeq.1993.1061
- Victor Zurkowski, Polynomial flows in the plane: a classification based on spectra of derivations, J. Differential Equations 120 (1995), no. 1, 1–29. MR 1339667, DOI 10.1006/jdeq.1995.1103
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- 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