# Multiplicity (mathematics)

*https://en.wikipedia.org/wiki/Multiplicity_(mathematics)*

In
mathematics, the **multiplicity** of a member of a
multiset is the number of times it appears in the multiset. For example, the number of times a given
polynomial has a
root at a given point is the multiplicity of that root.

The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, *double roots* counted twice). Hence the expression, "counted with multiplicity".

If multiplicity is ignored, this may be emphasized by counting the number of **distinct** elements, as in "the number of distinct roots". However, whenever a
set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".

## Multiplicity of a prime factor

In prime factorization, for example,

- 60 = 2 × 2 × 3 × 5,

the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors.

## Multiplicity of a root of a polynomial

Let be a
field and be a
polynomial in one variable with
coefficients in . An element is a **
root of multiplicity** of if there is a polynomial such that and . If , then *a* is called a **simple root**. If , then is called a **multiple root**.

For instance, the polynomial has 1 and −4 as roots, and can be written as . This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra.

If is a root of multiplicity of a polynomial, then it is a root of multiplicity of its derivative. The discriminant of a polynomial is zero if and only if the polynomial has a multiple root.

### Behavior of a polynomial function near a multiple root

The
graph of a
polynomial function *f* touches the *x*-axis at the real roots of the polynomial. The graph is
tangent to it at the multiple roots of *f* and not tangent at the simple roots. The graph crosses the *x*-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity.

A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an such that .

## Intersection multiplicity

In
algebraic geometry, the intersection of two sub-varieties of an algebraic variety is a finite union of
irreducible varieties. To each component of such an intersection is attached an **intersection multiplicity**. This notion is
local in the sense that it may be defined by looking at what occurs in a neighborhood of any
generic point of this component. It follows that without loss of generality, we may consider, in order to define the intersection multiplicity, the intersection of two
affines varieties (sub-varieties of an affine space).

Thus, given two affine varieties *V*_{1} and *V*_{2}, consider an
irreducible component *W* of the intersection of *V*_{1} and *V*_{2}. Let *d* be the
dimension of *W*, and *P* be any generic point of *W*. The intersection of *W* with *d*
hyperplanes in
general position passing through *P* has an irreducible component that is reduced to the single point *P*. Therefore, the
local ring at this component of the
coordinate ring of the intersection has only one
prime ideal, and is therefore an
Artinian ring. This ring is thus a
finite dimensional vector space over the ground field. Its dimension is the **intersection multiplicity** of *V*_{1} and *V*_{2} at *W*.

This definition allows us to state Bézout's theorem and its generalizations precisely.

This definition generalizes the multiplicity of a root of a polynomial in the following way. The roots of a polynomial *f* are points on the
affine line, which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is where *K* is an
algebraically closed field containing the coefficients of *f*. If is the factorization of *f*, then the local ring of *R* at the prime ideal is This is a vector space over *K*, which has the multiplicity of the root as a dimension.

This definition of intersection multiplicity, which is essentially due to
Jean-Pierre Serre in his book *Local Algebra*, works only for the set theoretic components (also called *isolated components*) of the intersection, not for the
embedded components. Theories have been developed for handling the embedded case (see
Intersection theory for details).

## In complex analysis

Let *z*_{0} be a root of a
holomorphic function *f*, and let *n* be the least positive integer such that, the *n*^{th} derivative of *f* evaluated at *z*_{0} differs from zero. Then the power series of *f* about *z*_{0} begins with the *n*^{th} term, and *f* is said to have a root of multiplicity (or “order”) *n*. If *n* = 1, the root is called a simple root.^{
[1]}

We can also define the multiplicity of the
zeroes and
poles of a
meromorphic function thus: If we have a meromorphic function , take the
Taylor expansions of *g* and *h* about a point *z*_{0}, and find the first non-zero term in each (denote the order of the terms *m* and *n* respectively). if *m* = *n*, then the point has non-zero value. If , then the point is a zero of multiplicity . If , then the point has a pole of multiplicity .

## References

**^**(Krantz 1999, p. 70)

- Krantz, S. G.
*Handbook of Complex Variables*. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.