The Invariance of Milnor's Number Implies the Invariance of the Topological Type

Topological type in families of germs

“…The following corollary generalizes both [5] and [12] since for complex analytic germs, no coalescing means exactly that Milnor's number Ft is constant. …”

“…In the complex case there is almost a complete result. La and Ramanujam in [5] combined with [2] have shown that if fz: (C",0)-,(C,0) is a family of germs of analytic functions with no coalescing and n ~ 3 then the topological type of each fz is the same. (The case n= 3 is unknown).…”

Invariants of hypersurface singularities in positive characteristic

“…It is a non-trivial theorem, using methods from complex analysis, which cannot be extended to other fields that the Milnor number is indeed invariant under contact equivalence (see [6, p. 262]), and it is even a topological invariant (see [15]). Using the Lefschetz Principle the result for contact equivalence can be generalised to arbitrary algebraically closed fields of characteristic zero (see [2, Prop.…”

Redundant Picard–Fuchs System for Abelian Integrals

“…The uniqueness assumption means that all these n 2 points coincide at the origin. By the famous theorem due to D. T. Le^and C. P. Ramanujam [19], the topological type of an analytic germ is constant along the stratum +=const, therefore the germs of H 0 and H 1 at the origin are topologically equivalent, in particular, the germs of analytic curves [H 0 =0] and [H 1 =0] in (C 2 , 0) are homeomorphic. But by the Zariski theorem [27], the order of a planar analytic curve (i.e., the order of the lowest order terms which occur in the Taylor expansion of the local equation defining this curve) is a topological invariant.…”