Multiplicity of a local ring

Summary

The multiplicity of a Noetherian local ring $(R, \mathfrak m)$ of dimension $d$ is the positive integer $e(R)$ such that the Hilbert–Samuel function $\ell(R/\mathfrak m^n)$ grows like $e(R) n^d / d!$. It measures how singular the corresponding point is. A regular local ring has multiplicity $1$, and a plane curve singularity has multiplicity equal to the lowest degree of a term in its equation.