The main focus of the theory is to detect and measure a singular point of an algebraic variety (cf.
Let R be a positively graded ring such that R is finitely generated as an R0-algebra and R0 is Artinian.
By definition, the multiplicity of M is The series may be rewritten where r(t) is a polynomial.
The following theorem, due to Christer Lech, gives a priori bounds for multiplicity.
[1][2] Lech — Suppose R is local with maximal ideal