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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the polynomial algebra in two variables over a field of characteristic . A subalgebra of is called a retract if there is an idempotent homomorphism (a *retraction*, or *projection*) such that 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 up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.

**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**

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