Parusinski’s “Key Lemma” via algebraic geometry

Authors:
Z. Reichstein and B. Youssin

Journal:
Electron. Res. Announc. Amer. Math. Soc. **5** (1999), 136-145

MSC (1991):
Primary 14E15, 14F10, 14L30; Secondary 16S35, 32B10, 58A40

DOI:
https://doi.org/10.1090/S1079-6762-99-00072-4

Published electronically:
November 17, 1999

MathSciNet review:
1728678

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Abstract | References | Similar Articles | Additional Information

Abstract: The following “Key Lemma” plays an important role in the work by Parusiński 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 $\mathbb {C}^n$. We give a new algebro-geometric proof of this result.

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Additional Information

**Z. Reichstein**

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

MR Author ID:
268803

**B. Youssin**

Affiliation:
Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel

Address at time of publication:
Hashofar 26/3, Ma’ale Adumim, Israel

Email:
youssin@math.bgu.ac.il

Received by editor(s):
October 16, 1999

Published electronically:
November 17, 1999

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

Communicated by:
David Kazhdan

Article copyright:
© Copyright 1999
American Mathematical Society