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 (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

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 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)^ks(x). If k = 1, then a is called a simple 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). Multiplicity can be thought of as "How many times does the solution appear in the original equation?".

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

Behavior of a polynomial function near a root in relation to its multiplicity

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 touch but not cross 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 be tangent to the x-axis, in other words it will have slope 0 there.

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 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 / x²) 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 z₀ 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 z₀ differs from zero. Then the power series of ƒ about z₀ 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 (Krantz 1999, p. 70).

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 z₀, and find the first non-zero term in each (denote the term numbers 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.

See also

• Zero (complex analysis)
• Set (mathematics)
• Fundamental theorem of algebra
• Fundamental theorem of arithmetic
• Algebraic multiplicity and geometric multiplicity of an eigenvalue
• Frequency (statistics)

References

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