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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.