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The Samuel stratification of the discriminant is Whitney regular. (English) Zbl 0557.57010

Let A denote the space of unitary polynomials \(x^ n+a_ 1x^{n- 1}+...+a_ n\), \(a_ i\in {\mathbb{C}}\) of degree n. The discriminant \(D\subset A\) is the algebraic hypersurface consisting of those polynomials having a multiple root. The Samuel stratification of the discriminant is the partition of D into the subsets \(D^ m=\{a\in D\); \(mult_ a(D)=m\}\) of constant multiplicity. It is shown that this Samuel stratification coincides with the canonical Whitney stratification of the algebraic set D. The proof is based on the connection between the discriminant D and the versal deformation of the fat point \(X_ 0:x^ n=0\) of type \(A_{n-1}\).

MSC:

57N80 Stratifications in topological manifolds
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
58A35 Stratified sets
30D10 Representations of entire functions of one complex variable by series and integrals
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