Parusinski’s “Key Lemma” via algebraic geometry

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 n , there is a finite set of homogeneous symmetric polynomials W 1 , … , W N W_1, \dots ,W_N in Z [ x 1 , … , x n ] Z[x_1,\dots ,x_n] and a constant M > 0 M >0 such that \[ | d x i / x i | ≤ M max j = 1 , … , N | d W j / W j | , |dx_i/x_i| \le M \max _{j = 1, \dots , N} |dW_j/W_j| \; , \] as densely defined functions on the tangent bundle of C n \mathbb {C}^n . We give a new algebro-geometric proof of this result.