Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.
The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice).
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".
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.
of the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of k, in which case
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
There is also a Macaulay dual space of differential functionals at
The dimension of this Macaulay dual space is the multiplicity of the solution
The Macaulay dual space forms the multiplicity structure of the system at the solution.
with is of multiplicity 3 because the Macaulay dual space is of dimension 3, where
The multiplicity is always finite if the solution is isolated, is perturbation invariant in the sense that a
under perturbation in complex spaces, and is identical to the intersection multiplicity on polynomial systems.
In algebraic geometry, the intersection of two sub-varieties of an algebraic variety is a finite union of irreducible varieties.
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 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 hyperplanes 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
where K is an algebraically closed field containing the coefficients of f. If
is the factorization of f, then the local ring of R at the prime ideal
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.
Let z0 be a root of a holomorphic function f, and let n be the least positive integer such that the nth derivative of f evaluated at z0 differs from zero.
[4] We can also define the multiplicity of the zeroes and poles of a meromorphic function.
take the Taylor expansions 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.
