Weierstrass preparation theorem

The following theorem is known as the Weierstrass preparation theorem, though sometimes that name is reserved for the corollary and this theorem is then known as the Weierstrass division theorem.

In the following we use the standard notation for coordinatesPlanetmathPlanetmath in n that z=(z1,,zn)=(z,zn). That is z is the first n-1 coordinates.


Let f:CnC be a function analyticPlanetmathPlanetmath in a neighbourhood U of the origin such that f(0,zn)znm extends to be analytic at the origin and is not zero at the origin for some positive integer m (in other words, as a function of zn, the function has a zero of order m at the origin). Then there exists a polydisc DU such that every function g holomorphic and boundedPlanetmathPlanetmath in D can be written as


where q is an analytic function and r is a polynomial in the zn variable of degree less then m with the coefficients being holomorphic functions in z. Further there exists a constant C independent of g such that


The representation g=qf+r is unique. Finally the coefficients of the power seriesMathworldPlanetmath expansions of q and r are finite linear combinationsMathworldPlanetmath of the coefficients of the power series of g.

Note that r is not necessarily a Weierstrass polynomial.


Let f be as above, then there is a unique representation of f as f=hW, where h is analytic in a neighbourhood of the origin and h(0)0 and W being a Weierstrass polynomial.

It should be noted that the condition that f(0,zn)znm extends to be analytic, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that f(0,zn)0, is not an essential restrictionPlanetmathPlanetmath. In fact f(0,zn)0, then there exists a linear change of coordinates, arbitrarily close to the identityPlanetmathPlanetmathPlanetmath, such that the condition of the theorem is satisfied in the new set of coordinates.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Weierstrass preparation theorem
Canonical name WeierstrassPreparationTheorem
Date of creation 2013-03-22 15:04:28
Last modified on 2013-03-22 15:04:28
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Theorem
Classification msc 32B05
Synonym Weierstrass division theorem
Related topic WeierstrassPolynomial