Logarithmic vector fields for the discriminants of composite functions

The $K_f$-equivalence is a natural equivalence between map-germs $\varphi\mathbb C^m\mathbb C^n$ which ensures that their compositions $f\circ\varphi$ with a fixed function-germ f on $\mathbb C^n$ are the same up to biholomorphisms of $\mathbb C^m$. We show that the discriminant $\sum$ in the base of a $K_f$-versal deformation of a germ $\varphi$ is Saito's free divisor provided the critical locus of f is Cohen–Macaulay of codimension $m+1$ and all the transversal types of $f$ are $A_k$ singularities. We give an algorithm to construct basic vector fields tangent to $\sum$. This is a generalisation of classical Zakalyukin's algorithm to write out basic fields tangent to the discriminant of an isolated function singularity. The case of symmetric matrix families in two variables is done in detail. For simple singularities, it is directly related to Arnold's convolution of invariants of Weyl groups.