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Parusinski's ``Key Lemma'' via algebraic geometry
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## Parusinski's ``Key Lemma'' via algebraic geometry

### Z. Reichstein and B. Youssin

**Abstract.**
The following ``Key Lemma'' plays an important role in the work by
Parusi\'nski
on the existence of Lipschitz stratifications in the class of
semianalytic sets:
For any positive integer $n$, there is a finite set of homogeneous
symmetric polynomials $W_1, \dots ,W_N$ in $Z[x_1,\dots,x_n]$
and a constant $M >0$ such that
\[ |dx_i/x_i| \le M \max_{j = 1, \dots, N} |dW_j/W_j| \; , \]
as densely defined functions on the tangent bundle of $\bbC^n$.
We give a new algebro-geometric proof of this result.

*Copyright 1999 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**05** (1999), pp. 136-145
- Publisher Identifier: S 1079-6762(99)00072-4
- 1991
*Mathematics Subject Classification*. Primary 14E15, 14F10, 14L30; Secondary 16S35, 32B10, 58A40
- Received by the editors October 16, 1999
- Posted on November 17, 1999
- Communicated by David Kazhdan
- Comments (When Available)

**Z. Reichstein**

Department of Mathematics, Oregon State University, Corvallis, OR 97331

*E-mail address:* `zinovy@math.orst.edu`

**B. Youssin**

Department of Mathematics and Computer Science, University of the Negev, Be'er Sheva', Israel

*Current address:* Hashofar 26/3, Ma'ale Adumim, Israel

*E-mail address:* `youssin@math.bgu.ac.il`

Z. Reichstein was partially supported by NSF grant DMS-9801675 and (during his stay at MSRI) by NSF grant DMS-9701755.

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