On the Rigidity of Some Extensions of Domains

A ring is said to be rigid if it admits no nonzero locally nilpotent derivations, and an affine variety is rigid if its coordinate ring is rigid. In this paper, we improve some techniques for determin-ing the rigidity of k-domains (affine varieties) over a field k of char-acteristic zero. First, we generalize the ABC theorem. Then we study locally nilpotent derivations of a simple algebraic extension R[z] of a k-domain R, where rzn e R for some nonzero r e R and some pos-itive integer n. Subsequently, we study locally nilpotent derivations and rigidity of an extension R[x, y] of R such that r1xmyn e R or r1xm +r2yn e R for some nonzero r1, r2 e R and some positive in-tegers m, n. Finally, as applications of these general results, we prove the rigidity of some quadrinomial varieties and Pham-Brieskorn hy-persurfaces.