Lower Bound of Newton Number

We show a lower estimate of the Milnor number of an isolated hypersurface singularity, via its Newton number. We also obtain analogous estimate of the Milnor number of an isolated singularity of a similar complete intersection variety. Introduction We study the Newton number of a polyhedron in order to calculate the Milnor number of an isolated singularity defined by an analytic mapping. Section 2 treats the Newton number of a quasi-convenient polyhedron. In Section 3, we consider the Milnor number μ(f, 0) of an isolated hypersurface singular point 0 ∈ C defined by a function f ∈ C{z1, · · · , zn}. It is well known that the Milnor number of a critical point 0 of a semi-quasihomogeneous function f is equal to that of its initial part ([MO]-Thm.1, [A]-Thm.3.1, [LR]-Cor.2.4). On the other hand, Kouchnirenko proved that μ(f, 0) ≥ ν(f) holds for any function f ∈ C[[z1, · · · , zn]], where ν(f) is the Newton number of f ([K]). We show a lower estimate of μ(f, 0) for not necessarily semiquasihomogeneous function f as follows, μ(f, 0) ≥ ν(g) ≥ (a1 − 1) · · · · · (an − 1), where (a1, 0, · · · , 0), · · · , (0, · · · , 0, an) are vertices of an arbitrary n− 1 dimensional simplex lying below Γ(f) with ai ≥ 1 (i = 1, · · · , n), and g is a standard modification of f to a convenient function. When all the ai are integers, this result follows from Kouchnirenko’s Theorem and upper semicontinuity of μ under a deformation. Recently Tomari and I have given a simple proof of this theorem which is quite different from the one of this note ([TF]). In Section 4, we mention a μ-constant family of an isolated hypersurface singularity. In Section 5, we consider the case of complete intersection singularities. Oka obtained a formula of the Milnor number of an isolated similar complete intersection singularity ([O2]) which is a generalization of Kouchnirenko’s formula ([K]). We also obtain a lower estimate of the Milnor number of an isolated similar complete intersection singularity. running title. Newton number 2000 Mathematics Subject Classification. Primary 14B05; Secondary 32S05.