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, the multiplicity of a prime factor is its p {\displaystyle p} -adic valuation. For example, the prime factorization of the integer 60 is
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 {\displaystyle F} be a field and p ( x ) {\displaystyle p(x)} be a polynomial in one variable with coefficients in F {\displaystyle F} . An element a ∈ F {\displaystyle a\in F} is a root of multiplicity k {\displaystyle k} of p ( x ) {\displaystyle p(x)} if there is a polynomial s ( x ) {\displaystyle s(x)} such that s ( a ) ≠ 0 {\displaystyle s(a)\neq 0} and p ( x ) = ( x − a ) k s ( x ) {\displaystyle p(x)=(x-a)^{k}s(x)} . If k = 1 {\displaystyle k=1} , then a is called a simple root. If k ≥ 2 {\displaystyle k\geq 2} , then a {\displaystyle a} is called a multiple root. For example, the polynomial p ( x ) = x 3 + 2 x 2 − 7 x + 4 {\displaystyle p(x)=x^{3}+2x^{2}-7x+4} has 1 and −4 as roots and can be written as p ( x ) = ( x + 4 ) ( x − 1 ) 2 {\displaystyle p(x)=(x+4)(x-1)^{2}} , which 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 {\displaystyle a} is a root of multiplicity k {\displaystyle k} of a polynomial, then it is a root of multiplicity k − 1 {\displaystyle k-1} of the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of k, in which case a {\displaystyle a} is a root of multiplicity at least k {\displaystyle k} of the 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 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 traverses the x-axis at roots of odd multiplicity and does not traverse it at roots of even multiplicity, where "to traverse the x-axis" means that, near the root, there are points of the graph on both sides of the axis. A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an x 0 {\displaystyle x_{0}} such that f ( x 0 ) > 0 {\displaystyle f(x_{0})>0} .
Multiplicity of a solution of a nonlinear system of equations For an equation f ( x ) = 0 {\displaystyle f(x)=0} with a single variable solution x ∗ {\displaystyle x_{*}} , the multiplicity is k {\displaystyle k} if
f ( x ∗ ) = f ′ ( x ∗ ) = ⋯ = f ( k − 1 ) ( x ∗ ) = 0 {\displaystyle f(x_{*})=f'(x_{*})=\cdots =f^{(k-1)}(x_{*})=0} and f ( k ) ( x ∗ ) ≠ 0. {\displaystyle f^{(k)}(x_{*})\neq 0.}
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