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• Weierstrass_preparation_theorem abstract "In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients are analytic functions in the remaining variables and zero at P.There are also a number of variants of the theorem, that extend the idea of factorization in some ring R as u·w, where u is a unit and w is some sort of distinguished Weierstrass polynomial. C.L. Siegel has disputed the attribution of the theorem to Weierstrass, saying that it occurred under the current name in some of late nineteenth century Traités d'analyse without justification.For one variable, the local form of an analytic function f(z) near 0 is zkh(z) where h(0) is not 0, and k is the order of zero of f at 0. This is the result the preparation theorem generalises. We pick out one variable z, which we may assume is first, and write our complex variables as (z, z2, ..., zn). A Weierstrass polynomial W(z) is zk + gk−1zk−1 + ... + g0where gi(z2, ..., zn)is analytic and gi(0, ..., 0) = 0.Then the theorem states that for analytic functions f, if f(0, ...,0) = 0,but f(z, z2, ..., zn)as a power series has some term only involving z, we can write (locally near (0, ..., 0)) f(z, z2, ..., zn) = W(z)h(z, z2, ..., zn)with h analytic and h(0, ..., 0) not 0, and W a Weierstrass polynomial.This has the immediate consequence that the set of zeros of f, near (0, ..., 0), can be found by fixing any small values of z2, ..., zn and then solving the equation W(z)=0. The corresponding values of z form a number of continuously-varying branches, in number equal to the degree of W in z. In particular f cannot have an isolated zero.A related result is the Weierstrass division theorem, which states that if f and g are analytic functions, and g is a Weierstrass polynomial of degree N, then there exists a unique pair h and j such that f = gh + j, where j is a polynomial of degree less than N. This is equivalent to the preparation theorem, since the Weierstrass factorization of f may be obtained by applying the division theorem for g = zN for the least N that gives an h not zero at the origin; the desired Weierstrass polynomial is then zN + j/h. For the other direction, we use the preparation theorem on g, and the normal polynomial remainder theorem on the resulting Weierstrass polynomial.There is a deeper preparation theorem for smooth functions, due to Bernard Malgrange, called the Malgrange preparation theorem. It also has an associated division theorem, named after John Mather.".
• Weierstrass_preparation_theorem wikiPageID "372198".
• Weierstrass_preparation_theorem wikiPageRevisionID "572400726".
• Weierstrass_preparation_theorem first "E.D.".
• Weierstrass_preparation_theorem hasPhotoCollection Weierstrass_preparation_theorem.
• Weierstrass_preparation_theorem id "W/w097510".
• Weierstrass_preparation_theorem last "Solomentsev".
• Weierstrass_preparation_theorem title "Weierstrass theorem".
• Weierstrass_preparation_theorem subject Category:Commutative_algebra.
• Weierstrass_preparation_theorem subject Category:Several_complex_variables.
• Weierstrass_preparation_theorem subject Category:Theorems_in_complex_analysis.
• Weierstrass_preparation_theorem type Abstraction100002137.
• Weierstrass_preparation_theorem type Communication100033020.
• Weierstrass_preparation_theorem type Message106598915.
• Weierstrass_preparation_theorem type Proposition106750804.
• Weierstrass_preparation_theorem type Statement106722453.
• Weierstrass_preparation_theorem type Theorem106752293.
• Weierstrass_preparation_theorem type TheoremsInComplexAnalysis.
• Weierstrass_preparation_theorem comment "In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P.".
• Weierstrass_preparation_theorem label "Théorème de préparation de Weierstrass".
• Weierstrass_preparation_theorem label "Weierstrass preparation theorem".
• Weierstrass_preparation_theorem sameAs Théorème_de_préparation_de_Weierstrass.
• Weierstrass_preparation_theorem sameAs m.020l8c.
• Weierstrass_preparation_theorem sameAs Q3527219.
• Weierstrass_preparation_theorem sameAs Q3527219.
• Weierstrass_preparation_theorem sameAs Weierstrass_preparation_theorem.
• Weierstrass_preparation_theorem wasDerivedFrom Weierstrass_preparation_theorem?oldid=572400726.
• Weierstrass_preparation_theorem isPrimaryTopicOf Weierstrass_preparation_theorem.