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: 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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- Adam Parusiński, Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 4, 189–213 (English, with French summary). MR 978246
- 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.
- 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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- S. Montgomery, Fixed points of finite automorphism groups of associative rings, Lect. Notes in Math. 818, Springer-Verlag, 1980.
- D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Third enlarged edition, Springer, 1994.
- A. Parusiński, Lipschitz properties of semianalytic sets, Ann. Inst. Fourier, Grenoble 38 (1988), 189–213.
- 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.
- I. R. Shafarevich, Basic algebraic geometry, Springer-Verlag, Heidelberg, 1974.
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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
