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

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 Parusinski on the existence of Lipschitz stratifications in the class of semianalytic sets: For any positive integer , there is a finite set of homogeneous symmetric polynomials in and a constant such that

as densely defined functions on the tangent bundle of . We give a new algebro-geometric proof of this result.

**[B]**N. Bourbaki,*Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples*, Actualités Sci. Ind. no. 1261, Hermann, Paris, 1958 (French). MR**0098114****[Mo]**Susan Montgomery,*Fixed rings of finite automorphism groups of associative rings*, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR**590245****[MFK]**D. Mumford, J. Fogarty, and F. Kirwan,*Geometric invariant theory*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR**1304906****[P]**Adam Parusiński,*Lipschitz properties of semi-analytic sets*, Ann. Inst. Fourier (Grenoble)**38**(1988), no. 4, 189–213 (English, with French summary). MR**978246****[RY]**Z. Reichstein and B. Youssin,*Essential dimensions of algebraic groups and a resolution theorem for -varieties*, with an appendix by J. Kollár and E. Szabó, preprint. Available at http://ucs.orst.edu/reichstz/pub.html.**[Sh]**I. R. Shafarevich,*Basic algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR**0366917**

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

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