JournalscmhVol. 79 , No. 2DOI 10.1007/s00014-003-0796-3

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
The stable equivalence and cancellation problems cover

Abstract

Let K be an arbitrary field of characteristic 0, and An\mathbf{A}^n the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties V1,V2AnV_1, V_2 \subseteq \mathbf{A}^n with isomorphic cylinders V1×A1V_1 \times \mathbf{A}^1 and V2×A1V_2 \times \mathbf{A}^1, whether V1V_1 and V2V_2 themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of An+1\mathbf{A}^{n+1}) cylinders V1×A1V_1 \times \mathbf{A}^1 and V2×A1V_2 \times \mathbf{A}^1, are V1V_1 and V2V_2 equivalent under an automorphism of An\mathbf{A}^n? We call this stable equivalence problem. We show that the answer is positive for any two curves V1,V2A2V_1, V_2 \subseteq \mathbf{A}^2. For an arbitrary n2n \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.