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Inmathematics, themultiplicity of a member of amultiset is the number of times it appears in the multiset. For example, the number of times a givenpolynomial has aroot 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 ofdistinct elements, as in "the number of distinct roots". However, whenever aset (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".

Inprime factorization, themultiplicity of a prime factor is its${\displaystyle p}$-adic valuation. For example, the prime factorization of theinteger60 is

60 = 2 × 2 × 3 × 5,

the multiplicity of the prime factor2 is2, while the multiplicity of each of the prime factors3 and5 is1. Thus,60 has four prime factors allowing for multiplicities, but only three distinct prime factors.

Let${\displaystyle F}$ be afield and${\displaystyle p(x)}$ be apolynomial in one variable withcoefficients in${\displaystyle F}$. An element${\displaystyle a\in F}$ is aroot of multiplicity${\displaystyle k}$ of${\displaystyle p(x)}$ if there is a polynomial${\displaystyle s(x)}$ such that${\displaystyle s(a)\ne 0}$ and${\displaystyle p(x)=(x-a{)}^{k}s(x)}$. If${\displaystyle k=1}$, thena is called asimple root. If${\displaystyle k\ge 2}$, then${\displaystyle a}$ is called amultiple root.

For instance, the polynomial${\displaystyle p(x)=x^3+2x^2-7x+4}$ has 1 and −4 asroots, and can be written as${\displaystyle p(x)=(x+4)(x-1{)}^2}$. 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 thefundamental theorem of algebra.

If${\displaystyle a}$ is a root of multiplicity${\displaystyle k}$ of a polynomial, then it is a root of multiplicity${\displaystyle k-1}$ of thederivative of that polynomial, unless thecharacteristic of the underlying field is a divisor ofk, in which case${\displaystyle a}$ is a root of multiplicity at least${\displaystyle k}$ of the derivative.

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

Thegraph of apolynomial functionf touches thex-axis at the real roots of the polynomial. The graph istangent to it at the multiple roots off and not tangent at the simple roots. The graph crosses thex-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity.

A non-zero polynomial function is everywherenon-negative if and only if all its roots have even multiplicity and there exists an${\displaystyle x_0}$ such that${\displaystyle f(x_0)>0}$.

For an equation${\displaystyle f(x)=0}$ with a single variable solution${\displaystyle x_{\ast }}$, the multiplicity is${\displaystyle k}$ if

${\displaystyle f(x_{\ast })=f^{\prime }(x_{\ast })=\cdots =f^{(k-1)}(x_{\ast })=0}$ and${\displaystyle f^{(k)}(x_{\ast })\ne 0.}$

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

Let${\displaystyle \mathbf{f}(\mathbf{x})=\mathbf{0}}$ be a system of${\displaystyle m}$ equations of${\displaystyle n}$ variables with a solution${\displaystyle {\mathbf{x}}_{\ast }}$ where${\displaystyle \mathbf{f}}$ is a mapping from${\displaystyle R^n}$ to${\displaystyle R^m}$ or from${\displaystyle C^n}$ to${\displaystyle C^m}$. There is also a Macaulay dual space of differential functionals at${\displaystyle {\mathbf{x}}_{\ast }}$ in which every functional vanishes at${\displaystyle \mathbf{f}}$. The dimension of this Macaulay dual space is the multiplicity of the solution${\displaystyle {\mathbf{x}}_{\ast }}$ to the equation${\displaystyle \mathbf{f}(\mathbf{x})=\mathbf{0}}$. The Macaulay dual space forms the multiplicity structure of the system at the solution.^[2][3]

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

${\displaystyle \mathbf{f}(\mathbf{x})=\left[\begin{array}{c}\sin (x_1)-x_2+x_1^2\\ x_1-\sin (x_2)+x_2^2\end{array}\right]}$

is of multiplicity 3 because the Macaulay dual space

${\displaystyle \mathrm{span}\{{\mathrm{\partial }}_{00},{\mathrm{\partial }}_{10}+{\mathrm{\partial }}_{01},-{\mathrm{\partial }}_{10}+{\mathrm{\partial }}_{20}+{\mathrm{\partial }}_{11}+{\mathrm{\partial }}_{02}\}}$

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

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

Inalgebraic geometry, the intersection of two sub-varieties of an algebraic variety is a finite union ofirreducible varieties. To each component of such an intersection is attached anintersection multiplicity. This notion islocal in the sense that it may be defined by looking at what occurs in a neighborhood of anygeneric point of this component. It follows that without loss of generality, we may consider, in order to define the intersection multiplicity, the intersection of twoaffines varieties (sub-varieties of an affine space).

Thus, given two affine varietiesV₁ andV₂, consider anirreducible componentW of the intersection ofV₁ andV₂. Letd be thedimension ofW, andP be any generic point ofW. The intersection ofW withdhyperplanes ingeneral position passing throughP has an irreducible component that is reduced to the single pointP. Therefore, thelocal ring at this component of thecoordinate ring of the intersection has only oneprime ideal, and is therefore anArtinian ring. This ring is thus afinite dimensional vector space over the ground field. Its dimension is the intersection multiplicity ofV₁ andV₂ atW.

This definition allows us to stateBézout'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 polynomialf are points on theaffine line, which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is${\displaystyle R=K[X]/⟨f⟩,}$ whereK is analgebraically closed field containing the coefficients off. If${\displaystyle f(X)=\prod _{i=1}^k(X-{\alpha }_i{)}^{m_i}}$ is the factorization off, then the local ring ofR at the prime ideal${\displaystyle ⟨X-{\alpha }_i⟩}$ is${\displaystyle K[X]/⟨(X-\alpha {)}^{m_i}⟩.}$ This is a vector space overK, which has the multiplicity${\displaystyle m_i}$ of the root as a dimension.

This definition of intersection multiplicity, which is essentially due toJean-Pierre Serre in his bookLocal Algebra, works only for the set theoretic components (also calledisolated components) of the intersection, not for theembedded components. Theories have been developed for handling the embedded case (seeIntersection theory for details).

Letz₀ be a root of aholomorphic functionf, and letn be the least positive integer such that then^th derivative off evaluated atz₀ differs from zero. Then thepower series off aboutz₀ begins with then^th term, andf is said to have a root of multiplicity (or “order”) n. Ifn = 1, the root is called a simple root.^[4]

We can also define the multiplicity of thezeroes andpoles of ameromorphic function. If we have a meromorphic function$f=\frac{g}{h},$ take theTaylor expansions ofg andh about a pointz₀, and find the first non-zero term in each (denote the order of the termsm andn respectively) then ifm = n, then the point has non-zero value. If${\displaystyle m>n,}$ then the point is a zero of multiplicity${\displaystyle m-n.}$ If, then the point has a pole of multiplicity${\displaystyle n-m.}$

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

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