# On the rigidity of certain Pham-Brieskorn rings

## TL;DR

This paper investigates the rigidity of Pham-Brieskorn rings, a class of algebraic structures, providing partial results towards a conjecture about their properties when certain conditions on exponents are met.

## Contribution

It offers new partial results and insights into the conjecture regarding the rigidity of Pham-Brieskorn rings under specific conditions.

## Key findings

- Partial confirmation of the rigidity conjecture for certain cases.
- Identification of conditions under which Pham-Brieskorn rings are rigid.
- Advancement in understanding the structure of these algebraic 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.