On the Zariski multiplicity conjecture for weighted homogeneous and Newton nondegenerate line singularities

Abstract

We present new families of weighted homogeneous and Newton non-degenerate line singularities that satisfy the Zariski multiplicity conjecture. * In this paper, we are only interested in the behaviour of functions (or hypersurfaces) near the origin 0 ∈ C n , unless otherwise stated. Hereafter, we shall omit the words "at 0." † It is well known that if the family { f t } is topologically equisingular, then the Milnor number µ( f t ) is independent of t for all small t. Both conditions are equivalent if n = 3 (cf. [16]).Remark. Combined with A.2, it follows that (14) ⇒ (4) if n = 4.