# 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 equation has a root at a given point.

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 the 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 4 prime factors, but only 3 distinct prime factors.

## Multiplicity of a root of a polynomial

Let *F* be a field and *p*(*x*) be a polynomial in one variable and coefficients in *F*. An element *a* ∈ *F* is called a root of multiplicity *k* of *p*(*x*) if there is a polynomial *s*(*x*) such that *s*(*a*) ≠ 0 and *p*(*x*) = (*x* − *a*)^{k}*s*(*x*). If *k* = 1, then *a* is called a simple root. If *k* ≥ 2, then *a* is called a **multiple root**.

For instance, the polynomial *p*(*x*) = *x*^{3} + 2*x*^{2} − 7*x* + 4 has 1 and −4 as roots, and can be written as *p*(*x*) = (*x* + 4)(*x* − 1)^{2}. This means that 1 is a root of multiplicity 2, and −4 is a 'simple' root (of multiplicity 1). Multiplicity can be thought of as "How many times does the solution appear in the original equation?".

The derivative of a polynomial has a multiplicity *n* - 1 at a root of multiplicity *n* of the polynomial.
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 *y* = *f*(*x*) intersects the *x*-axis at the real roots of the polynomial. The graph is tangent to this axis 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 bounces off (not goes through) the *x*-axis at roots of even multiplicity.

A non-zero polynomial function is always non-negative if and only if all its roots have an even multiplicity and there exists *x*_{0} such that *f*(*x*_{0}) > 0.

## 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 what occurs in a neighborhood of any generic point of this component. It follows that without loss of generality, we may consider, for defining 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}, let us 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 to state precisely Bézout's theorem and its generalizations.

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 * ƒ *, and let *n* be the least positive integer such that, the *n*^{th} derivative of *ƒ* evaluated at *z*_{0} differs from zero. Then the power series of *ƒ* about *z*_{0} begins with the *n*^{th} term, and *ƒ* 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 *ƒ* = *g*/*h*, 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 *m* > *n*, then the point is a zero of multiplicity *m* − *n*. If *m* < *n*, then the point has a pole of multiplicity *n* − *m*.

## References

- ↑ (Krantz 1999, p. 70)

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