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

Author(s): Z. Reichstein; B. Youssin
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 136-145.
MSC (1991): Primary 14E15, 14F10, 14L30; Secondary 16S35, 32B10, 58A40
Posted: November 17, 1999
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Abstract: The following ``Key Lemma'' plays an important role in the work by Parusinski 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

\begin{displaymath}|dx_i/x_i| \le M \max _{j = 1, \dots, N} |dW_j/W_j| \; , \end{displaymath}

as densely defined functions on the tangent bundle of ${\mathbb C}^n$. We give a new algebro-geometric proof of this result.

References:

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N. Bourbaki, Algèbre, Hermann, Paris. MR 20:4576

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S. Montgomery, Fixed points of finite automorphism groups of associative rings, Lect. Notes in Math. 818, Springer-Verlag, 1980. MR 81j:16041

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D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Third enlarged edition, Springer, 1994. MR 95m:14012

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A. Parusi\'{n}ski, Lipschitz properties of semianalytic sets, Ann. Inst. Fourier, Grenoble 38 (1988), 189-213. MR 90e:32016
[RY]
Z. Reichstein and B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for $G$-varieties, with an appendix by J. Kollár and E. Szabó, preprint. Available at http://ucs.orst.edu/$\tilde{\;}$reichstz/pub.html.

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I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, Heidelberg, 1974. MR 51:3163

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Additional Information:

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

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

DOI: 10.1090/S1079-6762-99-00072-4
PII: S 1079-6762(99)00072-4
Received by editor(s): October 16, 1999
Posted: 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
Copyright of article: Copyright 1999, American Mathematical Society

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