Multiplicity

In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. For example, the term is used to refer to the number of times a given polynomial equation has a root at a given point.

The common reason to consider notions of multiplicity is to count correctly, without specifying exceptions (for example, double roots counted twice). Hence the expression counted with (sometimes implicit) multiplicity.

When mathematicians wish to ignore multiplicity they will refer to the number of distinct elements of a set.

Multiplicity of a prime factor
In the prime factorization


 * 60 = 2 &times; 2 &times; 3 &times; 5

the multiplicity of the prime factor 2 is 2, while the multiplicity 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 &minus; a)ks(x). If k = 1, then a is called a simple root.

For instance, the polynomial p(x) = x3 + 2x2 &minus; 7x + 4 has 1 and &minus;4 as roots, and can be written as p(x) = (x + 4)(x &minus; 1)2. This means that 1 is a root of multiplicity 2, and &minus;4 is a 'simple' root (of multiplicity 1).

The discriminant of a polynomial is zero if and only if the polynomial has a multiple root.

Geometric behavior
Let f(x) be a polynomial function. Then, if f is graphed on a Cartesian coordinate system, its graph will cross the x-axis at real zeros of odd multiplicity and will 'bounce' from the x-axis at real zeros of even multiplicity. In addition, if f(x) has a zero with a multiplicity greater than 1, the graph will appear flatter at values close to it.

Multiplicity of a zero of a function
Let $$I$$ be an interval of R, let $$f$$ be a function from $$I$$ into R or C be a real (resp. complex) function, and let $$c$$ ∈ $$I$$ be a zero of $$f$$, i.e. a point such that $$f(c)=0$$. The point $$c$$ is said a zero of multiplicity $$k$$ of $$f$$ if there exist a real number $$\ell\neq 0$$ such that


 * $$\lim_{x\to c}\frac{|f(x)|}{|x-c|^k}=\ell.$$

In a more general setting, let $$f$$ be a function from an open subset $$A$$ of a normed vector space $$E$$ into a normed vector space $$F$$, and let $$c \in A$$ be a zero of $$f$$, i.e. a point such that $$f(c) = 0$$. The point $$c$$ is said a zero of multiplicity $$k$$ of $$f$$ if there exist a real number $$\ell \neq 0$$ such that


 * $$\lim_{x\to c}\frac{\|f(x)\|_{\mathcal F}}{\|x-c\|_{\mathcal E}^k}=\ell.$$

The point $$c$$ is said a zero of multiplicity ∞ of $$f$$ if for each $$k$$, it holds that


 * $$\lim_{x\to c}\frac{\|f(x)\|_{\mathcal F}}{\|x-c\|_{\mathcal E}^k}=0.$$

Example 1. Since


 * $$\lim_{x\to 0}\frac{|\sin x|}{|x|}=1,$$

0 is a zero of multiplicity 1 for the function sine function.

Example 2. Since


 * $$\lim_{x\to 0}\frac{|1-\cos x|}{|x|^2}=\frac 12,$$

0 is a zero of multiplicity 2 for the function $$1-\cos$$.

Example 3. Consider the function $$f$$ from R into R such that $$f(0) = 0$$ and that $$f(x)= \exp(1/x^2)$$ when $$x \neq 0$$. Then, since


 * $$\lim_{x\to 0}\frac{|f(x)|}{|x|^k}=0 \mbox{ for each }k \in \mathbb{N}$$

0 is a zero of multiplicity ∞ for the function $$f$$.

In complex analysis
Let $$z_0$$ be a root of a holomorphic function $$f$$, and let $$n$$ be the least positive integer $$m$$ such that, the $$m$$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 (Krantz 1999, p. 70).