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  2. Bézout's theorem - Wikipedia

    en.wikipedia.org/wiki/Bézout's_theorem

    This proves Bézout's theorem, if the multiplicity of a common zero is defined as the multiplicity of the corresponding linear factor of the U-resultant. As for the preceding proof, the equality of this multiplicity with the definition by deformation results from the continuity of the U -resultant as a function of the coefficients of the f i ...

  3. Hilbert's Nullstellensatz - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_Nullstellensatz

    In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields.

  4. Root-finding algorithm - Wikipedia

    en.wikipedia.org/wiki/Root-finding_algorithm

    In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f ( x ) = 0 . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form , root-finding algorithms provide approximations to zeros.

  5. Multiplicity (mathematics) - Wikipedia

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

    We can also define the multiplicity of the zeroes and poles of a meromorphic function. If we have a meromorphic function =, take the Taylor expansions of g and h about a point z 0, 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.

  6. Nine windows - Wikipedia

    en.wikipedia.org/wiki/Nine_windows

    The nine windows technique, also known as 9 windows, 9 boxes, 9 screens, multiscreen diagram, or system operator tool is a creative problem-solving technique that analyzes a problem across time and relative to its place within a system. [1] [2] [3] [4]

  7. Newton's method - Wikipedia

    en.wikipedia.org/wiki/Newton's_method

    If the multiplicity m of the root is finite then g(x) = ⁠ f(x) / f ′ (x) ⁠ will have a root at the same location with multiplicity 1. Applying Newton's method to find the root of g(x) recovers quadratic convergence in many cases although it generally involves the second derivative of f(x).

  8. Intersection number - Wikipedia

    en.wikipedia.org/wiki/Intersection_number

    Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:

  9. Particular values of the Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Particular_values_of_the...

    The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.