##  [Multiplicity](/index.php/multiplicity) 

  ##  [Multiplicity](https://puremath.quantumdictionary.io/multiplicity-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/index.php/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A numerical invariant (often Hilbert–Samuel multiplicity) that measures the leading coefficient of a Hilbert polynomial or the asymptotic growth rate of lengths of M/I^nM for an m-primary ideal I in a local ring; it quantifies local algebraic concentration or singularity order.

 

 

 

 

 





 

 



 ##  [Multiplicity](https://algebra.quantumdictionary.io/multiplicity-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/index.php/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

Multiplicity is the integer count describing how many times a given root, eigenvalue, or intersection occurs: algebraic (root) multiplicity is the exponent of a factor (x−α)^m in a polynomial, and algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial; geometric multiplicity counts independent eigenvectors.