In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
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 exampl ...
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 su ...
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
(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
-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 languag ...
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
be a
field and
be a
polynomial
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 exampl ...
in one variable with
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s in
. An element
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 su ...
of multiplicity
of
if there is a polynomial
such that
and
. If
, then ''a'' is called a simple root. If
, then
is called a multiple root.
For instance, the polynomial
has 1 and −4 as
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
, and can be written as
. 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
is a root of multiplicity
of a polynomial, then it is a root of multiplicity
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
is a root of multiplicity at least
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 orig ...
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
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 exa ...
''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. Mo ...
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
such that
.
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 geometrical ...
, 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 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
affines varieties (sub-varieties of an affine space).
Thus, given two affine varieties ''V''
1 and ''V''
2, consider an
irreducible component ''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 coord ...
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 hyper ...
s in
general position 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 algebrai ...
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
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
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 rela ...
, which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is
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
is the factorization of ''f'', then the local ring of ''R'' at the prime ideal
is
This is a vector space over ''K'', which has the multiplicity
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 components. Theories have been developed for handling the embedded case (see
Intersection theory 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 deriv ...
''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 ...
of a
meromorphic function. If we have a meromorphic function
take the
Taylor expansions
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
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
then the point is a zero of multiplicity
If