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 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

Main article: p-adic order

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 four prime factors allowing for multiplicities, but only three 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 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)^ks(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) = x³ + 2x² − 7x + 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). 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 a is a root of multiplicity k of a polynomial, then it is a root of multiplicity k – 1 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

Graph of the polynomial p(x) = x³ + 2x² − 7x + 4 with its roots (zeros) −4 and 1. The root −4 is a 'simple' root (of multiplicity 1), and therefore the graph crosses the x-axis at this root. The root 1 is of even multiplicity and therefore the graph bounces off the x-axis at this 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 x₀ such that f(x₀) > 0.

Intersection multiplicity

Main article: Intersection theory

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 V₁ and V₂, let us consider an irreducible component W of the intersection of V₁ and V₂. 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 V₁ and V₂ 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 $R=K[X]/\langle f\rangle ,$ where K is an algebraically closed field containing the coefficients of f. If $f(X)=\prod _{i=1}^{k}(X-\alpha _{i})^{m_{i}}$ is the factorization of f, then the local ring of R at the prime ideal $\langle X-\alpha _{i}\rangle$ is $K[X]/\langle (X-\alpha )^{m_{i}}\rangle .$ This is a vector space over K, which has the multiplicity $m_{i}$ 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 z₀ 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 z₀ differs from zero. Then the power series of f about z₀ 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.

We can also define the multiplicity of the zeroes and poles of a meromorphic function thus: If we have a meromorphic function f = g/h, take the Taylor expansions of g and h about a point z₀, 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, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.