# Weierstrass polynomial

###### Definition.

A function $W\colon{\mathbb{C}}^{n}\to{\mathbb{C}}$ of the form

 $W(z_{1},\ldots,z_{n})=z_{n}^{m}+\sum_{j=1}^{m-1}a_{j}(z_{1},\ldots,z_{n-1})z_{% n}^{j},$

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.

## References

• 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 polynomial WeierstrassPolynomial 2013-03-22 15:04:25 2013-03-22 15:04:25 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 32A17 msc 32B05 W-polynomial Multifunction WeierstrassPreparationTheorem