Multiplicity is a general term meaning "the number of values for which a given condition holds." For example, the term is used to refer to the value of the totient valence function or the number of times a given polynomial equation has a root at a given point.
Let 
be a root
of a function 
, and let 
be the least positive integer 
such that 
. Then the power
series of 
about 
begins with the 
th
term,
 |
and 
is said to have a root
of multiplicity (or "order") 
. If 
, the root is called a simple root (Krantz 1999, p. 70).
For a reduced hypersurface germ 
at the origin in 
, multiplicity is the least
total degree of a nonzero term of the convergent
power series 
. For instance, 
has multiplicity 2 at the origin. The Zariski
multiplicity conjecture asks whether this integer
is determined by the embedded topological germ.
