What kind of singularity is commonly meant by $A\_{\infty}$?



$A_{\infty}$ singularity
For $\mu \ge 0,$ the $A\_{\mu}$ series of map germs $f : (\mathbb{R}^n,0) \to (\mathbb{R},0)$ is, for $\varepsilon\_i = \pm 1$, given by $f(x) = \varepsilon\_1x\_1^2 + \cdots + \varepsilon\_{n-1}x\_{n-1}^2 \pm x\_n^{\mu +1}$. These all have algebraically isolated singularities at $0 \in \mathbb{R}^n$. I this setting, finite Milnor number is equivalent to the singularity being isolated. For $\mu = \infty$, the $A\_{\infty}$ singularity is non-isolated, and is given by $f(x) = \varepsilon\_1x\_1^2 + \cdots + \varepsilon\_{n-1}x\_{n-1}^2$. Notice the lack of $x\_n$. If you're working over $\mathbb{C}$ then drop all of the $\pm$s.


