Multiplicities of Maxwell sets of Pham singularities
Let \(f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)\) be an isolated singularity and \(F:(\mathbb{C}^{n}\times \mathbb{C}^{k},0)\rightarrow ( \mathbb{C},0)\) a versal deformation of \(f.\) The \textit{caustic} (resp. \textit{Maxwell set}) of \(F\) is the set (resp. closure of the set) of all parameters \(\lambda \in \mathbb{C}^{k}\) such that the corresponding function \( f_{\lambda }=F(\cdot ,\lambda )\) has a non-Morse critical point (resp. has equal critical values at different critical points) close to the origin in \( \mathbb{C}^{n}.\) These sets are analytic. Denote by \(C(f)\) and \(M(f)\) the multiplicities of these sets at \(0\in \mathbb{C}^{k}\). The results are: 1. if \(f=z_{1}^{a_{1}+1}+\cdots +z_{n}^{a_{n}+1}\) is a Pham singularity then \[ \begin{aligned} C(f)& =(a_{1}\cdots a_{n})\left( \frac{a_{1}-1}{a_{1}}+\cdots +\frac{a_{n}-1 }{a_{n}}\right) , \\ 3C(f)+2M(f)& \leq \sum_{i=1}^{n}(a_{1}\cdots a_{i-1})\left( a_{i}^{2}-1\right) (a_{i+1}\cdots a_{n})^{2}, \end{aligned} \] 2. if additionally \(a_{1}=\cdots =a_{n}=0\) then \[ M(f)=\frac{a^{n-1}\left( \left( a+1\right) \left( a^{n}-1\right) -3n\left( a-1\right) \right) }{2}, \] 3. if \(f\) is an arbitrary isolated singularity then \[ 3C(f)+2M(f)\geq \frac{\mu (f)\left( \mu (f)-1\right) d}{d-1}, \] where \(\mu (f)\) is the Milnor number of \(f\) and \(d\) the order of \(f\) at \(0.\)
scientific article; zbMATH DE number 6681944