A zero has a «multiplicity», which refers to the frequency at which the associated factor appears in the polynomial. For example, squaring (x + 3)(x − 2) has zeros x = −3 and x = 2, which occur once. The eleventh degree polynomial (x + 3)4(x − 2)7 has the same zeros as the square, but in this case the solution x = − 3 has a multiplicity 4 because the factor (x + 3) occurs four times (i.e. the factor is increased to the fourth power) and the solution x = 2 has a multiplicity 7 because the factor (x − 2) occurs seven times. Let z0 be the root of a holomorphic function f and n the least positive integer, so that the nth derivative of f evaluated at z0 is nonzero. Then the power series of f starts around z0 with the nth term, and f is said to have a multiplicity root (or «order») n. If n = 1, the root is called a simple root. [1] x = −5 with multiplicity 3x = −2 with multiplicity 4x = 1 with multiplicity 2x = 5 with multiplicity 1 In civil proceedings, the term «multiplicity of actions» or «multiplication of actions» refers to the bringing of several actions that raise the same problem and that could have been brought in a single action. The law usually tries to avoid various measures, as this could lead to inconsistent results. Moreover, the admission of multiple unnecessary actions is «prejudicial to the legitimate interest of society in the efficiency of justice.

The courts are a public resource for publicly funded private dispute resolution. The next zero occurs at [latex]x=-1[/latex]. The graphs seem almost linear at this point. It is a single zero of multiplicity 1. This definition generalizes the multiplicity of a root of a polynomial as follows. 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 R = K [ X ] / ⟨ f ⟩ , {displaystyle R=K[X]/langle frangle ,}, where K is an algebraically closed field containing the coefficients of f. If f ( X ) = ∏ i = 1 k ( X − α i ) m i {displaystyle f(X)=prod _{i=1}^{k}(X-alpha _{i})^{m_{i}}} is the factorization of f, then the local ring of R at the prime ideal is ⟨ X − α i ⟩ {displaystyle langle X-alpha _{i}rangle } is K [ X ] / ⟨ ( X − α ) m i ⟩. {displaystyle K[X]/langle (X-alpha )^{m_{i}}rangle .} It is a vector space over K that has the multiplicity m i {displaystyle m_{i}} of the root as dimension. Let F {displaystyle F} be a field and p (x) {displaystyle p(x)} be a polynomial in a variable with coefficients in F {displaystyle F}.

An element a ∈ F {displaystyle ain F} is a root of the multiplicity k {displaystyle k} of p ( x ) {displaystyle p(x)} if there exists a polynomial s ( x ) {displaystyle s(x)} such that s ( a ≠) 0 {displaystyle s(a)neq 0} and p ( x ) = ( x − a ) k s ( x ) {displaystyle p(x)=(x-a)^{k}s(x)}. If k = 1 {displaystyle k=1} , then a is called a simple root. If k 2 {displaystyle kgeq 2} ≥, then a {displaystyle a} is called a multiple root. Thus, for two affine manifolds V1 and V2, consider an irreducible component W of the intersection of V1 and V2. Let d be the dimension of W and P any generic point of W. The intersection of W with d hyperplanes in general position passing through P has an irreducible component 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 artinic ring. This ring is therefore a finite-dimensional vector space over the fundamental field. Its dimension is the intersection of V1 and V2 to W. If multiplicity is ignored, this can be evidenced by counting the number of different elements, as in «the number of different roots».

However, when a set (as opposed to multiset) is formed, multiplicity is automatically ignored without the need to use the term «distinct». If a {displaystyle a} is a root of the multiplicity k {displaystyle k} of a polynomial, then it is a root of the multiplicity k − 1 {displaystyle k-1} of the derivative of that polynomial, unless the property of the underlying field is a divisor of k, in which case a {displaystyle a} is a root of the multiplicity at least k {displaystyle k} of the derivative. This definition of intersection, which is essentially due to Jean-Pierre Serre in his book Local Algebra, only works for the components of set theory (also called isolated components) of the intersection, not for the embedded components. Theories have been developed to deal with the integrated case (see Intersection theory for more details). n. several actual or potential claims that should be combined into a single lawsuit and suit. One of the fundamental principles of the law is that diversity should be avoided as much as possible, achievable and equitable. Example: Several lawsuits are filed by different people against the same person or organization, based on the same facts and legal issues. At the request of one of the parties or by decision of the judge, the judge may order the consolidation of cases.

In algebraic geometry, the intersection of two subvarieties of an algebraic variety is a finite union of irreducible varieties. An intersection is attached to each element of such a passage. This term is local in the sense that it can be defined by looking at what happens in a neighborhood of a generic point of this component. It follows that, without losing the generality, for the definition of intersection, one can consider the intersection of two affine manifolds (submanifolds of an affine space). The point of multiplicity with respect to the graph is that all factors that occur an even number of times (i.e. all zeros that occur twice, four times, six times, etc.) are squares, so they do not change the sign. Squares are always positive. This means that the intersection x that corresponds to a zero of even multiplicity cannot cross the x-axis because zero cannot change the graph from positive sign (above the x-axis) to negative (below the x-axis) or vice versa. 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».