On the rigidity of certain Pham-Brieskorn rings

Abstract

Fix a field $k$ of characteristic zero. If $a_1,...,a_n$ ($n>2$) are positive integers, the integral domain $B=k[X_1,...,X_n]/(X_1^{a_1}+...+X_n^{a_n})$ is called a Pham-Brieskorn ring. It is conjectured that if $a_i>1$ for all $i$ and $a_i=2$ for at most one $i$, then $B$ is rigid. (A ring $B$ is said to be rigid if the only locally nilpotent derivation $D:B\to B$ is the zero derivation.) We give partial results towards the conjecture.