|
|
|
|
Weierstrass polynomial
|
(Definition)
|
|
Definition 1 A function $W\colon {\mathbb{C}}^n \to {\mathbb{C}}$ of the form \begin{equation*} W(z_1,\ldots,z_n) = z_n^m + \sum_{j=1}^{m-1}a_j(z_1,\ldots,z_{n-1})z_n^j , \end{equation*}where the $a_j$ are holomorphic functions in a neighbourhood of the origin, which vanish at the origin, is called a Weierstrass polynomial.
Any codimension 1 complex analytic subvariety of ${\mathbb{C}}^n$ can be written as the zero set of a Weierstrass polynomial using the Weierstrass preparation theorem. This in general cannot be done for higher codimension.
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|
"Weierstrass polynomial" is owned by jirka.
|
|
(view preamble | get metadata)
Cross-references: Weierstrass preparation theorem, zero set, complex analytic subvariety, codimension, vanish, origin, neighbourhood, holomorphic functions, function
There are 3 references to this entry.
This is version 4 of Weierstrass polynomial, born on 2005-02-22, modified 2007-12-18.
Object id is 6795, canonical name is WeierstrassPolynomial.
Accessed 2382 times total.
Classification:
| AMS MSC: | 32B05 (Several complex variables and analytic spaces :: Local analytic geometry :: Analytic algebras and generalizations, preparation theorems) | | | 32A17 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Special families of functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
