The Samuel stratification of the discriminant is Whitney regular (Q761667)

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 \(A_{n-1}\).