Cancellation problem via locally nilpotent derivations

Abstract

The Zariski cancellation problem plays a central role in affine algebraic geometry and noncommutative algebra, with locally nilpotent derivations providing a fundamental invariant-theoretic approach. This article presents a unified survey of cancellation phenomena in commutative algebras, noncommutative algebras, and skew (Ore-type) extensions, emphasizing the role of rigidity and the Makar--Limanov invariant. We explain how the locally nilpotent derivation framework successfully detects cancellation in rigid settings, while also identifying its inherent limitations, particularly in the skew case where Makar--Limanov stability fails. This perspective clarifies the scope and the boundaries of the locally nilpotent derivation method in cancellation theory.