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ISSN 1079-6762

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 $n$, 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 [Enhancements On Off] (What's this?)

  • [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 (20 #4576)
  • [Mo] Susan Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Mathematics, vol. 818, Springer, Berlin, 1980. MR 590245 (81j:16041)
  • [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 (95m:14012)
  • [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 (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$\tilde{\;}$reichstz/pub.html.
  • [Sh] I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, New York, 1974. Translated from the Russian by K. A. Hirsch; Die Grundlehren der mathematischen Wissenschaften, Band 213. MR 0366917 (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

PII: S 1079-6762(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