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Parusinski’s “Key Lemma” via algebraic geometry. (English) Zbl 0956.32009
Summary: 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 \[ |dx_i/x_i|\leq M \max_{j = 1, \dots, N} |dW_j/W_j|, \] as densely defined functions on the tangent bundle of \({\mathbb C}^n\). We give a new algebro-geometric proof of this result.

32B20 Semi-analytic sets, subanalytic sets, and generalizations
14P15 Real-analytic and semi-analytic sets
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