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  2. Multiplicity (mathematics) - Wikipedia

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

    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.

  3. Geometrical properties of polynomial roots - Wikipedia

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

    It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjugation: the roots of a polynomial with rational coefficients are conjugate (that is, invariant) under the action of the Galois group of the polynomial. However, this symmetry can rarely be ...

  4. Polynomial - Wikipedia

    en.wikipedia.org/wiki/Polynomial

    The number of roots of a nonzero polynomial P, counted with their respective multiplicities, cannot exceed the degree of P, [25] and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra). The coefficients of a polynomial and its roots are related by Vieta's formulas.

  5. Fundamental theorem of algebra - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

    The n complex numbers , …, are the roots of the polynomial. If a root appears in several factors, it is a multiple root, and the number of its occurrences is, by definition, the multiplicity of the root.

  6. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    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.

  7. Minimal polynomial (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Minimal_polynomial_(linear...

    The multiplicity of a root λ of μ A is the largest power m such that ker((A − λI n) m) strictly contains ker((A − λI n) m−1). In other words, increasing the exponent up to m will give ever larger kernels , but further increasing the exponent beyond m will just give the same kernel.

  8. Algebraic curve - Wikipedia

    en.wikipedia.org/wiki/Algebraic_curve

    An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.

  9. Bézout's theorem - Wikipedia

    en.wikipedia.org/wiki/Bézout's_theorem

    This definition of a multiplicities by deformation was sufficient until the end of the 19th century, but has several problems that led to more convenient modern definitions: Deformations are difficult to manipulate; for example, in the case of a root of a univariate polynomial, for proving that the multiplicity obtained by deformation equals ...