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In mathematics, the multiplicity of a member of a
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
is the number of times it appears in the multiset. For example, the number of times a given
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
has a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
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 In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...
, the multiplicity of a prime factor is its p-adic valuation. For example, the prime factorization of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
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.


Multiplicity of a root of a polynomial

Let F be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
and p(x) be a
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
in one variable with coefficients in F. An element a \in F is a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of multiplicity k of p(x) if there is a polynomial s(x) such that s(a)\neq 0 and p(x) = (x-a)^k s(x). If k=1, then ''a'' is called a simple root. If k \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 The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynom ...
. If a is a root of multiplicity k of a polynomial, then it is a root of multiplicity k-1 of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
of a
polynomial function In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
''f'' touches the ''x''-axis at the real roots of the polynomial. The graph is
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mor ...
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 In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or i ...
if and only if all its roots have even multiplicity and there exists an x_0 such that f(x_0) > 0.


Intersection multiplicity

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, 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 Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administra ...
in the sense that it may be defined by looking at what occurs in a neighborhood of any
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
of this component. It follows that without loss of generality, we may consider, in order to define the intersection multiplicity, the intersection of two affines varieties (sub-varieties of an affine space). Thus, given two affine varieties ''V''1 and ''V''2, consider an
irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (fo ...
''W'' of the intersection of ''V''1 and ''V''2. Let ''d'' be the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
of ''W'', and ''P'' be any generic point of ''W''. The intersection of ''W'' with ''d''
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
s in
general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ...
passing through ''P'' has an irreducible component that is reduced to the single point ''P''. Therefore, the
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic nu ...
at this component of the
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
of the intersection has only one
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
, 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''1 and ''V''2 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 In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
, which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is R=K \langle f\rangle, where ''K'' is an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
containing the coefficients of ''f''. If f(X)=\prod_^k (X-\alpha_i)^ is the factorization of ''f'', then the local ring of ''R'' at the prime ideal \langle X-\alpha_i\rangle is K \langle (X-\alpha)^\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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the i ...
in his book ''Local Algebra'', works only for the set theoretic components (also called ''isolated components'') of the intersection, not for the embedded components. Theories have been developed for handling the embedded case (see
Intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem ...
for details).


In complex analysis

Let ''z''0 be a root of a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriva ...
''f'', and let ''n'' be the least positive integer such that, the ''n''th derivative of ''f'' evaluated at ''z''0 differs from zero. Then the power series of ''f'' about ''z''0 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 Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Cen ...
of a
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), pole ...
. If we have a meromorphic function f = \frac, take the Taylor expansions of ''g'' and ''h'' about a point ''z''0, 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 m, then the point has a pole of multiplicity n-m.


References

*Krantz, S. G. ''Handbook of Complex Variables''. Boston, MA: Birkhäuser, 1999. {{isbn, 0-8176-4011-8. Set theory Mathematical analysis