# On simple $Z_2$-invariant and corner function germs

## TL;DR

This paper extends Arnold's classification of simple singularities to include $Z_2$-invariant and corner function germs, providing criteria based on intersection forms and monodromy groups.

## Contribution

It generalizes Arnold's results to arbitrary $Z_2$ actions and corner singularities, establishing new classification criteria.

## Key findings

- Extended classification criteria for $Z_2$-invariant germs.
- Established conditions involving intersection forms and monodromy groups.
- Proved analogues of Arnold's theorems for corner singularities.

## Abstract

V.I.Arnold has classified simple (i.e. having no modules for the classification) singularities (function germs), and also simple boundary singularities (function germs invariant with respect to the action $\sigma(x_1; y_1, \ldots, y_n)=(-x_1; y_1, \ldots, y_n)$ of the group $Z_2$. In particular, it was shown that a function germ (respectively a boundary singularity germ) is simple if and only if the intersection form (respectively the restriction of the intersection form to the subspace to anti-invariant cycles) of a germ in $3+4s$ variables stable equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding subspace is finite. We formulate and prove analogues of these statements for function germs invariant with respect to an arbitrary action of the group $Z_2$ and also for corner singularities.