Commentarii Mathematici Helvetici


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Volume 79, Issue 2, 2004, pp. 341–349
DOI: 10.1007/s00014-003-0796-3

The stable equivalence and cancellation problems

Leonid Makar-Limanov[1], Peter van Rossum[2], Vladimir Shpilrain[3] and JIE-TAI YU[4]

(1) Department of Mathematics, Wayne State University, MI 48202, DETROIT, UNITED STATES
(2) Department of Mathematical Sciences, New Mexico State University, NM 88011, LAS CRUCES, UNITED STATES
(3) Department of Mathematics, City University of New York, NY 10031, NEW YORK, UNITED STATES
(4) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HONG KONG, POKFULAM ROAD, HONG KONG, CHINA

Let K be an arbitrary field of characteristic 0, and $\mathbf{A}^n$ the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties $V_1, V_2 \subseteq \mathbf{A}^n$ with isomorphic cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, whether $V_1$ and $V_2$ themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of $\mathbf{A}^{n+1}$) cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, are $V_1$ and $V_2$ equivalent under an automorphism of $\mathbf{A}^n$? We call this stable equivalence problem. We show that the answer is positive for any two curves $V_1, V_2 \subseteq \mathbf{A}^2$. For an arbitrary $n \ge 2$, we consider a special, arguably the most important, case of both problems, where one of the varieties is a hyperplane. We show that a positive solution of the stable equivalence problem in this case implies a positive solution of the cancellation problem.

Keywords: Algebraic varieties, cancellation problem, polynomial automorphisms, stable equivalence, Danielewski surfaces

Makar-Limanov L, van Rossum P, Shpilrain V, YU J. The stable equivalence and cancellation problems. Comment. Math. Helv. 79 (2004), 341-349. doi: 10.1007/s00014-003-0796-3