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2. Multiplicity (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Multiplicity_(mathematics)

   In 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 has a root at a given point is the multiplicity of that root.

3. Scheme (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Scheme_(mathematics)

   In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).

4. Multiset - Wikipedia

   en.wikipedia.org/wiki/Multiset

   As a consequence, an infinite number of multisets exist that contain only elements a and b, but vary in the multiplicities of their elements: The set {a, b} contains only elements a and b, each having multiplicity 1 when {a, b} is seen as a multiset. In the multiset {a, a, b}, the element a has multiplicity 2, and b has multiplicity 1.

5. Serre's multiplicity conjectures - Wikipedia

   en.wikipedia.org/wiki/Serre's_multiplicity...

   Serre proved this for all regular local rings. He established the following three properties when R is either of equal characteristic or of mixed characteristic and unramified (which in this case means that characteristic of the residue field is not an element of the square of the maximal ideal of the local ring), and conjectured that they hold in general.

6. Geometrical properties of polynomial roots - Wikipedia

   en.wikipedia.org/wiki/Geometrical_properties_of...

   Kahan discovered that polynomials with a particular set of multiplicities form what he called a pejorative manifold and proved that a multiple root is Lipschitz continuous if the perturbation maintains its multiplicity. This geometric property of multiple roots is crucial in numerical computation of multiple roots.

7. Fundamental theorem of algebra - Wikipedia

   en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

   The fundamental theorem of algebra, also called d'Alembert's theorem [1] or the d'Alembert–Gauss theorem, [2] states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.