A cancellation conjecture for free associative algebras

Drensky, Vesselin; Yu, Jie-Tai

doi:10.1090/S0002-9939-08-09111-9

A cancellation conjecture for free associative algebras
HTML articles powered by AMS MathViewer

by Vesselin Drensky and Jie-Tai Yu PDF

Proc. Amer. Math. Soc. 136 (2008), 3391-3394 Request permission

Abstract:

We develop a new method to deal with the Cancellation Conjecture of Zariski in different environments. We prove the conjecture for free associative algebras of rank two. We also produce a new proof of the conjecture for polynomial algebras of rank two over fields of zero characteristic.

References

Shreeram S. Abhyankar, William Heinzer, and Paul Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310–342. MR 306173, DOI 10.1016/0021-8693(72)90134-2
George M. Bergman, Centralizers in free associative algebras, Trans. Amer. Math. Soc. 137 (1969), 327–344. MR 236208, DOI 10.1090/S0002-9947-1969-0236208-5
P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619, DOI 10.1007/978-3-0348-8440-2
Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106–110. MR 531454
Shulim Kaliman and Mikhail Zaidenberg, Families of affine planes: the existence of a cylinder, Michigan Math. J. 49 (2001), no. 2, 353–367. MR 1852308, DOI 10.1307/mmj/1008719778
Hanspeter Kraft, Challenging problems on affine $n$-space, Astérisque 237 (1996), Exp. No. 802, 5, 295–317. Séminaire Bourbaki, Vol. 1994/95. MR 1423629
Leonid Makar-Limanov, Peter van Rossum, Vladimir Shpilrain, and Jie-Tai Yu, The stable equivalence and cancellation problems, Comment. Math. Helv. 79 (2004), no. 2, 341–349. MR 2059436, DOI 10.1007/s00014-003-0796-3
Alexander A. Mikhalev, Vladimir Shpilrain, and Jie-Tai Yu, Combinatorial methods, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 19, Springer-Verlag, New York, 2004. Free groups, polynomials, and free algebras. MR 2014326, DOI 10.1007/978-0-387-21724-6
Masayoshi Miyanishi, Some remarks on polynomial rings, Osaka Math. J. 10 (1973), 617–624. MR 337957
Masayoshi Miyanishi and Tohru Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ. 20 (1980), no. 1, 11–42. MR 564667, DOI 10.1215/kjm/1250522319
Peter Russell, On affine-ruled rational surfaces, Math. Ann. 255 (1981), no. 3, 287–302. MR 615851, DOI 10.1007/BF01450704
Ivan P. Shestakov and Ualbai U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004), no. 1, 181–196. MR 2015333, DOI 10.1090/S0894-0347-03-00438-7
Vladimir Shpilrain and Jie-Tai Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), no. 1, 295–307. MR 1900285, DOI 10.1006/jabr.2001.9124
Jie Tai Yu, On relations between Jacobians and minimal polynomials, Linear Algebra Appl. 221 (1995), 19–29. MR 1331786, DOI 10.1016/0024-3795(93)00220-T