The stable equivalence and cancellation problems

  • Leonid Makar-Limanov

    Wayne State University, Detroit, USA
  • Peter van Rossum

    New Mexico State University, Las Cruces, USA
  • Vladimir Shpilrain

    City University of New York, USA
  • Jie-Tai Yu

    University of Hong Kong, China


Let K be an arbitrary field of characteristic 0, and the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties with isomorphic cylinders and , whether and themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of ) cylinders and , are and equivalent under an automorphism of ? We call this stable equivalence problem. We show that the answer is positive for any two curves . For an arbitrary , 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.

Cite this article

Leonid Makar-Limanov, Peter van Rossum, Vladimir Shpilrain, Jie-Tai Yu, The stable equivalence and cancellation problems. Comment. Math. Helv. 79 (2004), no. 2, pp. 341–349

DOI 10.1007/S00014-003-0796-3