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| 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_of_a_function|root]] at a given point.
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| 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".
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| 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==
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| {{main|p-adic order}}
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| In the [[Integer factorization|prime factorization]], for example, | |
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| : 60 = 2 × 2 × 3 × 5
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| 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.
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| ==Multiplicity of a root of a polynomial==
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| <!-- [[Eigenvalue, eigenvector and eigenspace#Definitions]] links to this section. Change the link there if you change this header -->
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| Let ''F'' be a [[field (mathematics)|field]] and ''p''(''x'') be a [[polynomial]] in one variable and coefficients in ''F''. An element ''a'' ∈ ''F'' is called a [[root of a function|root]] of multiplicity ''k'' of ''p''(''x'') if there is a polynomial ''s''(''x'') such that ''s''(''a'') ≠ 0 and ''p''(''x'') = (''x'' − ''a'')<sup>''k''</sup>''s''(''x''). If ''k'' = 1, then ''a'' is called a ''simple root''.
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| For instance, the polynomial ''p''(''x'') = ''x''<sup>3</sup> + 2''x''<sup>2</sup> − 7''x'' + 4 has 1 and −4 as [[Root_of_a_function|roots]], and can be written as ''p''(''x'') = (''x'' + 4)(''x'' − 1)<sup>2</sup>. 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 [[Root_of_a_function|solution]] appear in the original equation?".
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| The [[Discriminant#Discriminant_of_a_polynomial|discriminant of a polynomial]] is zero only if the polynomial has a multiple root.
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| === Behavior of a polynomial function near a root in relation to its multiplicity ===
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| 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 [[Root_of_a_function|zeros]] of odd multiplicity and will bounce off (not go through) 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.
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| In general, a polynomial with an ''n''-fold root will have a [[derivative]] with an (''n''−1)-fold root at that point.
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| ==Intersection multiplicity==
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| {{main|Intersection theory}}
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| In [[algebraic geometry]], the intersection of two sub-varieties of an algebraic variety is a finite union of [[irreducible variety|irreducible varieties]]. To each component of such an intersection is attached an '''intersection multiplicity'''. This notion is [[local property|local]] in the sense that it may be defined by looking what occurs in a neighborhood of any [[generic point]] of this component. It follows that without loss of generality, we may consider, for defining the intersection multiplicity, the intersection of two [[affine variety|affines varieties]] (sub-varieties of an affine space).
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| Thus, given two affine varieties ''V''<sub>1</sub> and ''V''<sub>2</sub>, let us consider an [[irreducible component]] ''W'' of the intersection of ''V''<sub>1</sub> and ''V''<sub>2</sub>. Let ''d'' be the [[dimension of an algebraic variety|dimension]] of ''W'', and ''P'' be any generic point of ''W''. The intersection of ''W'' with ''d'' [[hyperplane]]s 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 ''V''<sub>1</sub> and ''V''<sub>2</sub> at ''W''.
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| This definition allows to state precisely [[Bézout's theorem]] and its generalizations.
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| 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 <math>R=K[X]/\langle f\rangle, </math> where ''K'' is an [[algebraically closed field]] containing the coefficients of ''f''. If <math>f(X)=\prod_{i=1}^k (X-\alpha_i)^{m_i}</math> is the factorization of ''f'', then the local ring of ''R'' at the prime ideal <math>\langle X-\alpha_i\rangle</math> is <math>K[X]/\langle (X-\alpha)^{m_i}\rangle.</math> This is a vector space over ''K'', which has the multiplicity <math>m_i</math> of the root as a dimension.
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| 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|embedded components]]. Theories have been developed for handling the embedded case (see [[intersection theory]] for details).
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| ==In complex analysis==
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| Let ''z''<sub>0</sub> be a root of a [[holomorphic function]] '' ƒ '', and let ''n'' be the least positive integer such that, the ''n''<sup>th</sup> derivative of ''ƒ'' evaluated at ''z''<sub>0</sub> differs from zero. Then the power series of ''ƒ'' about ''z''<sub>0</sub> begins with the ''n''<sup>th</sup> 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).
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| We can also define the multiplicity of the [[Zero (complex analysis)|zeroes]] and [[Pole (complex analysis)|poles]] of a [[meromorphic function]] thus: If we have a meromorphic function ''ƒ'' = ''g''/''h'', take the [[Taylor series|Taylor expansions]] of ''g'' and ''h'' about a point ''z''<sub>0</sub>, and find the first non-zero term in each (denote the order of the terms ''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''.
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| ==See also==
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| * [[Zero (complex analysis)]]
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| * [[Set (mathematics)]]
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| * [[Fundamental theorem of algebra]]
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| * [[Fundamental theorem of arithmetic]]
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| * Algebraic multiplicity and geometric multiplicity of an [[eigenvalue]]
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| * [[Frequency (statistics)]]
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| ==References==
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| *Krantz, S. G. ''Handbook of Complex Variables''. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.
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| [[Category:Set theory]]
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| [[Category:Mathematical analysis]]
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49 year old Systems Analyst Branstrom from Flin Flon, spends time with passions for example macram, Cheap Car Rentals In Naples From $19 and snorkeling. Has finished a great round the world journey that included going to the Lushan National Park.
Here is my homepage rental cars