Affine Varieties with Equivalent Cylinders

✍ Scribed by Vladimir Shpilrain; Jie-Tai Yu

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 117 KB
• Volume: 251
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.9124

✦ Synopsis

A well-known cancellation problem asks when, for two algebraic varieties

In this paper, we address a related problem: when the equivalence (under an automorphism of C n+1 ) of two cylinders V 1 × C and V 2 × C implies the equivalence of their bases V 1 and V 2 under an automorphism of C n . We concentrate here on hypersurfaces and show that this problem establishes a strong connection between the cancellation conjecture of Zariski and the embedding conjecture of Abhyankar and Sathaye. We settle the problem in the affirmative for a large class of polynomials. On the other hand, we give examples of equivalent cylinders with inequivalent bases. (Those cylinders, however, are not hypersurfaces.)

Another result of interest is that, for an arbitrary field K, the equivalence of two polynomials in m variables under an automorphism of K x 1

x n , n ≥ m, implies their equivalence under a tame automorphism of K x 1

x 2n .  2002

Elsevier Science (USA)