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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.
This extended multiplicity function is commonly called simply the multiplicity function, and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the indicator function of a subset , and shares some properties with it.
The notion of the multiplicity of a module is a generalization of the degree of a projective variety. By Serre's intersection formula, it is linked to an intersection multiplicity in the intersection theory. The main focus of the theory is to detect and measure a singular point of an algebraic variety (cf. resolution of singularities).
The algebraic multiplicity μ A (λ i) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ − λ i) k divides evenly that polynomial. [9] [25] [26] Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues.
The fundamental theorem of algebra, also called d'Alembert's theorem [1] or the d'Alembert–Gauss theorem, [2] ... counted with multiplicity, exactly n complex roots.
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field.
In 1965 Jean-Pierre Serre described how to find the multiplicity of each intersection point by methods of commutative algebra and homological algebra. [1] This connection between a geometric notion of intersection and a homological notion of a derived tensor product has been influential and led in particular to several homological conjectures ...
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain problems in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil 's initial definition of intersection numbers , around 1949, there had been a question of how to provide a more flexible and computable theory, which Serre ...
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