regular local ring

A regular local ring is a type of mathematical structure used in algebra and geometry to study solutions to polynomial equations near a specific point. Think of it as a way to examine the smallest neighborhood around that point, capturing local properties. The "regular" part means this neighborhood is well-behaved—specifically, its dimension matches the minimum number of parameters needed to describe nearby solutions. This concept helps mathematicians understand smoothness and singularities in shapes defined by equations, providing a precise way to analyze complex geometric objects at fine detail.