Multiplicity (mathematics)

 ( Set Theory ) 

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".

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Multiplicity of a prime factor In prime factorization, the multiplicity of a prime factor is its p-adic valuation. For example, the prime factorization of the integer is

the multiplicity of the prime factor is , while the multiplicity of each of the prime factors and is . Thus, has four prime factors allowing for multiplicities, but only three distinct prime factors.

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Multiplicity of a root of a polynomial Let $F$ be a field and $p(x)$ be a polynomial in one variable with in $F$. An element $a\; \backslash in\; F$ is a root of multiplicity $k$ of $p(x)$ if there is a polynomial $s(x)$ such that $s(a)\backslash neq\; 0$ and $p(x)\; =\; (x-a)^k\; s(x)$. If $k=1$, then a is called a simple root. If $k\; \backslash geq\; 2$, then $a$ is called a multiple root.

For instance, the polynomial $p(x)\; =\; x^3\; +\; 2x^2\; -\; 7x\; +\; 4$ has 1 and −4 as roots, and can be written as $p(x)\; =\; (x+4)(x-1)^2$. 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 the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of , in which case $a$ is a root of multiplicity at least $k$ of the derivative.

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

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Behavior of a polynomial function near a multiple root The graph of a polynomial function f touches the x-axis at the real roots of the polynomial. The graph is tangent to it 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 does not cross it at roots of even multiplicity.

A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an $x\_0$ such that $f(x\_0)\; >\; 0$.

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Multiplicity of a solution of a nonlinear system of equations For an equation $f(x)=0$ with a single variable solution $x\_*$, the multiplicity is $k$ if

$f(x\_*)=f\text{'}(x\_*)\; =\; \backslash cdots\; =\; f^\{(k-1)\}(x\_*)=0$ and $f^\{(k)\}(x\_*)\backslash neq0.$

In other words, the differential functional $\backslash partial\_j$, defined as the derivative $\backslash frac\{1\}\{j!\}\backslash frac\{d^j\}\{dx^j\}$ of a function at $x\_*$, vanishes at $f$ for $j$ up to $k-1$. Those differential functionals $\backslash partial\_0,\; \backslash partial\_1,\backslash cdots,\backslash partial\_\{k-1\}$ span a vector space, called the Macaulay dual space at $x\_*$, and its dimension is the multiplicity of $x\_*$ as a zero of $f$.

Let $\backslash mathbf\{f\}(\backslash mathbf\{x\})=\backslash mathbf\{0\}$ be a system of $m$ equations of $n$ variables with a solution $\backslash mathbf\{x\}\_*$ where $\backslash mathbf\{f\}$ is a mapping from $R^n$ to $R^m$ or from $C^n$ to $C^m$. There is also a Macaulay dual space of differential functionals at $\backslash mathbf\{x\}\_*$ in which every functional vanishes at $\backslash mathbf\{f\}$. The dimension of this Macaulay dual space is the multiplicity of the solution $\backslash mathbf\{x\}\_*$ to the equation $\backslash mathbf\{f\}(\backslash mathbf\{x\})=\backslash mathbf\{0\}$. The Macaulay dual space forms the multiplicity structure of the system at the solution.

For example, the solution $\backslash mathbf\{x\}\_*=(0,0)$ of the system of equations in the form of $\backslash mathbf\{f\}(\backslash mathbf\{x\})=\backslash mathbf\{0\}$ with

$\backslash mathbf\{f\}(\backslash mathbf\{x\})=\backslash left\backslash begin\{array\}\{c\}$

is of multiplicity 3 because the Macaulay dual space

$\backslash operatorname\{span\}\; \backslash \{\backslash partial\_\{00\},\; \backslash partial\_\{10\}+\backslash partial\_\{01\},\; -\backslash partial\_\{10\}+\backslash partial\_\{20\}+\backslash partial\_\{11\}+\backslash partial\_\{02\}\backslash \}$

is of dimension 3, where $\backslash partial\_\{ij\}$ denotes the differential functional $\backslash frac\{1\}\{i!j!\}\backslash frac\{\backslash partial^\{i+j\}\}\{\backslash partial\; x\_1^i\; \backslash ,\; \backslash partial\; x\_2^j\}$ applied on a function at the point $\backslash mathbf\{x\}\_*=(0,0)$.

The multiplicity is always finite if the solution is isolated, is perturbation invariant in the sense that a $k$-fold solution becomes a cluster of solutions with a combined multiplicity $k$ under perturbation in complex spaces, and is identical to the intersection multiplicity on polynomial systems.

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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 property 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 affine variety (sub-varieties of an affine space).

Thus, given two affine varieties V₁ and V₂, 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 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 $R=KX/\backslash langle\; f\backslash rangle,$ where K is an algebraically closed field containing the coefficients of f. If $f(X)=\backslash prod\_\{i=1\}^k\; (X-\backslash alpha\_i)^\{m\_i\}$ is the factorization of f, then the local ring of R at the prime ideal $\backslash langle\; X-\backslash alpha\_i\backslash rangle$ is $KX/\backslash langle\; (X-\backslash alpha)^\{m\_i\}\backslash rangle.$ 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 prime. Theories have been developed for handling the embedded case (see Intersection theory for details).

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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.(Krantz 1999, p. 70)

We can also define the multiplicity of the zeroes and poles of a meromorphic function. If we have a meromorphic function $f\; =\; \backslash frac\{g\}\{h\},$ take the Taylor series 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) then if m =  n, then the point has non-zero value. If $m>n,$ then the point is a zero of multiplicity $m-n.$ If

• Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, 1999. .

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