Affine varieties with equivalent cylinders

Simple item page
dc.creatorShpilrain, Vladimir
dc.creatorYu, Jie-Tai
dc.date2001-10-31
dc.date.accessioned2026-07-07T04:44:09Z
dc.date.available2026-07-07T04:44:09Z
dc.descriptionA well-known cancellation problem asks when, for two algebraic varieties $V_1, V_2 \subseteq {\bf C}^n$, the isomorphism of the cylinders $V_1 \times {\bf C}$ and $V_2 \times {\bf C}$ implies the isomorphism of $V_1$ and $V_2$. In this paper, we address a related problem: when the equivalence (under an automorphism of ${\bf C}^{n+1}$) of two cylinders $V_1 \times {\bf C}$ and $V_2 \times {\bf C}$ implies the equivalence of their bases $V_1$ and $V_2$ under an automorphism of ${\bf C}^n$? 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 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 $K[x_1,..., x_n], n \ge m,$ implies their equivalence under a tame automorphism of $K[x_1,..., x_{2n}]$.
dc.description12 pages
dc.identifierhttps://arxiv.org/abs/math/0110324
dc.identifierhttp://arxiv.org/abs/math/0110324
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/62524
dc.subjectAlgebraic Geometry
dc.subject14E09, 14E25
dc.titleAffine varieties with equivalent cylinders
dc.typetext

Files

Collections

arXiv