Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Polynomial Retracts
and the Jacobian Conjecture


Authors: Vladimir Shpilrain and Jie-Tai Yu
Journal: Trans. Amer. Math. Soc. 352 (2000), 477-484
MSC (1991): Primary 13B25, 13P10; Secondary 14E09, 16S10
DOI: https://doi.org/10.1090/S0002-9947-99-02251-5
Published electronically: September 21, 1999
MathSciNet review: 1487631
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a retraction, or projection) $\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R.$ The presence of other, equivalent, definitions of retracts provides several different methods of studying and applying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $ K[x, y]$ up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.


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

  • 1. S.S. Abhyankar, T.-T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166. MR 52:407
  • 2. S.S. Abhyankar, Lectures on expansion techniques in algebraic geometry, Notes by Balwant Singh. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 57. Tata Institute of Fundamental Research, Bombay, 1977. MR 80m:14016
  • 3. H. Appelgate, H. Onishi, The Jacobian conjecture in two variables, J. Pure Appl. Algebra 37 (1985), 215-227. MR 87b:14005
  • 4. E. Artal-Bartolo, P. Cassou-Nogues, I. Luengo Velasco, On polynomials whose fibers are irreducible with no critical points, Math. Ann. 299 (1994), 477-490. MR 95g:14016
  • 5. H. Bass, E. Connell and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287-330. MR 83k:14028
  • 6. P.M. Cohn, Free rings and their relations, Second edition, Academic Press, London, 1985. MR 87e:16006
  • 7. E. Connell, J. Zweibel, Subrings invariant under polynomial maps, Houston J. Math. 20 (1994), 175-185. MR 95g:13006
  • 8. D. Costa, Retracts of polynomial rings, J. Algebra 44 (1977), 492-502.
  • 9. A. van den Essen, H. Tutaj, A remark on the two-dimensional Jacobian conjecture, J. Pure Appl. Algebra 96 (1994), 19-22. MR 55:2876
  • 10. J. Gwozdziewicz, Injectivity on one line, Bull. Soc. Sci. Lett. Lodz Ser. Rech. Deform. 15 (1993), 59-60. MR 95f:14024
  • 11. E. Formanek, Observations about the Jacobian conjecture, Houston J. Math. 20 (1994), 369-380. MR 95f:14027
  • 12. S. Kaliman, On the Jacobian conjecture, Proc. Amer. Math. Soc. 117 (1993), 45-51. MR 93e:14017
  • 13. O. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306.
  • 14. H. Kraft, On a question of Yosef Stein, Automorphisms of affine spaces (Curacao, 1994), 225-229, Kluwer Acad. Publ., Dordrecht, 1995. MR 96i:14013
  • 15. J. Lang, Newton polygons of Jacobian pairs, J. Pure Appl. Algebra 72 (1991), 39-51. MR 92i:14012
  • 16. E.C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), no. 3, 255-263. MR 96m:20039

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13B25, 13P10, 14E09, 16S10

Retrieve articles in all journals with MSC (1991): 13B25, 13P10, 14E09, 16S10


Additional Information

Vladimir Shpilrain
Affiliation: Department of Mathematics, The City College of New York, New York, New York 10031
Email: shpil@groups.sci.ccny.cuny.edu

Jie-Tai Yu
Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
Email: yujt@hkusua.hku.hk

DOI: https://doi.org/10.1090/S0002-9947-99-02251-5
Received by editor(s): March 11, 1997
Received by editor(s) in revised form: August 20, 1997
Published electronically: September 21, 1999
Additional Notes: The second author’s research was partially supported by RGC Fundable Grant 344/024/0002
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society