Affine Varieties with Equivalent Cylinders

A well-known cancellation problem asks when, for two algebraic varieties V1, V2 Cn, the isomorphism of the cylinders V1 × C and V2 × C implies the isomorphism of V1 and V2. In this paper, we address a related problem: when the equivalence (under an automorphism of Cn + 1) of two cylinders V1 × C and V2 × C implies the equivalence of their bases V1 and V2 under an automorphism of Cn. 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[x1, …, xn], n ≥ m, implies their equivalence under a tame automorphism of K[x1, …, x2n].