Triangular derivations related to problems on affine $n$-space
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- by Arno van den Essen and Peter van Rossum PDF
- Proc. Amer. Math. Soc. 130 (2002), 1311-1322 Request permission
Abstract:
This paper studies the Cancellation Problem, the Embedding Problem, and the Linearization Problem. It shows how these problems can be related to a special class of locally nilpotent derivations.References
- Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166. MR 379502
- Shreeram S. Abhyankar, On the semigroup of a meromorphic curve. I, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 249–414. MR 578864
- Teruo Asanuma, Non-linearizable algebraic $k^\ast$-actions on affine spaces, Invent. Math. 138 (1999), no. 2, 281–306. MR 1720185, DOI 10.1007/s002220050379
- S. M. Bhatwadekar and A. Roy, Some results on embedding of a line in $3$-space, J. Algebra 142 (1991), no. 1, 101–109. MR 1125207, DOI 10.1016/0021-8693(91)90219-X
- P. C. Craighero, A result on $m$-flats in $\textbf {A}^n_k$, Rend. Sem. Mat. Univ. Padova 75 (1986), 39–46 (English, with Italian summary). MR 847656
- Daniel Daigle and Gene Freudenburg, A counterexample to Hilbert’s fourteenth problem in dimension $5$, J. Algebra 221 (1999), no. 2, 528–535. MR 1728394, DOI 10.1006/jabr.1999.7987
- James K. Deveney and David R. Finston, $G_a$ actions on $\textbf {C}^3$ and $\textbf {C}^7$, Comm. Algebra 22 (1994), no. 15, 6295–6302. MR 1303005, DOI 10.1080/00927879408825190
- 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, An algorithm to compute the invariant ring of a $\textbf {G}_a$-action on an affine variety, J. Symbolic Comput. 16 (1993), no. 6, 551–555. MR 1279532, DOI 10.1006/jsco.1993.1062
- A. van den Essen. Polynomial Automorphisms and the Jacobian Conjecture, volume 190 of Progress in Mathematics. Birkhäuser-Verlag, Basel–Boston–Berlin, 2000.
- G. Freudenburg. A counterexample to Hilbert’s fourteenth problem in dimension six. J. of Transformation Groups, 5:61–71, 2000.
- Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
- Zbigniew Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), no. 1, 113–120. MR 884649, DOI 10.1007/BF01457281
- Hanspeter Kraft, Challenging problems on affine $n$-space, Astérisque 237 (1996), Exp. No. 802, 5, 295–317. Séminaire Bourbaki, Vol. 1994/95. MR 1423629
- M. Miyanishi. Normal affine subalgebras of a polynomial ring. In Algebraic and Topological Theories, pages 37–51, Tokyo, 1985.
- A. Nowicki. Polynomial Derivations and their Rings of Constants. Univ. of Toruń, 1994.
- V. L. Popov, On actions of $\textbf {G}_a$ on $\textbf {A}^n$, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 237–242. MR 911143, DOI 10.1007/BFb0079241
- Rudolf Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A384–A387 (French). MR 232770
- W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
- V. Srinivas, On the embedding dimension of an affine variety, Math. Ann. 289 (1991), no. 1, 125–132. MR 1087241, DOI 10.1007/BF01446563
- Masakazu Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace $\textbf {C}^{2}$, J. Math. Soc. Japan 26 (1974), 241–257 (French). MR 338423, DOI 10.2969/jmsj/02620241
Additional Information
- Arno van den Essen
- Affiliation: Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands
- Email: essen@sci.kun.nl
- Peter van Rossum
- Affiliation: Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands
- Address at time of publication: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
- Email: petervr@sci.kun.nl, petervr@nmsu.edu
- Received by editor(s): May 24, 2000
- Received by editor(s) in revised form: November 12, 2000
- Published electronically: October 23, 2001
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1311-1322
- MSC (2000): Primary 13B25, 14E25
- DOI: https://doi.org/10.1090/S0002-9939-01-06212-8
- MathSciNet review: 1879952