On the real preimages of a real point under the Liashko–Looijenga covering for simple singularities

The multiplicity of the Lyashko–Looijenga covering restricted to a connected component of the complement in $\mathbf R^\mu$ of the real bifurcation diagram of functions with a simple singularity, is expressed in terms of some discrete invariants of this component. For the case of the singularities $A_\mu$ an algorithm is indicated for finding these invariants from the full system of invariants of the component proposed by S.  A. Barannikov.