Geodesic
Parusinski’s “Key Lemma” via algebraic geometry
Reichstein, Z.1 ; Youssin, B.2, 3
1Department of Mathematics, Oregon State University, Corvallis, OR 97331
2Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel
3Hashofar 26/3, Ma’ale Adumim, Israel
Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 136-145
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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.
DOI : 10.1090/S1079-6762-99-00072-4

Reichstein, Z.  1   ; Youssin, B.  2 , 3

1 Department of Mathematics, Oregon State University, Corvallis, OR 97331
2 Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel
3 Hashofar 26/3, Ma’ale Adumim, Israel 
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Reichstein, Z.; Youssin, B. Parusinski’s “Key Lemma” via algebraic geometry. Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 136-145. doi: 10.1090/S1079-6762-99-00072-4

[1] Bourbaki, N. Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples 1958

[2] Montgomery, Susan Fixed rings of finite automorphism groups of associative rings 1980

[3] Mumford, D., Fogarty, J., Kirwan, F. Geometric invariant theory 1994

[4] Parusiński, Adam Lipschitz properties of semi-analytic sets Ann. Inst. Fourier (Grenoble) 1988 189 213

[5] Shafarevich, I. R. Basic algebraic geometry 1974

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