Counting $D_4$ singularities in the image of a wave front

We give a formula to count the number of $D_4$ singularities in a stable frontal perturbation of a corank $2$ wave front singularity $f\text{\hspace{0.17em}⁣}:({\mathbb{C}}^3,0)\to ({\mathbb{C}}^4,0)$ using Mond’s method of stable perturbations of map germs. For a generic germ of corank $2$ wave front $f\text{\hspace{0.17em}⁣}:({\mathbb{C}}^3,S)\to ({\mathbb{C}}^4,0)$, the image of a stable deformation $f_t$ of $f$ exhibits $A_k$ singularities with $k\le 4$, their transverse intersections and the aforementioned $D_4$ singularities for $0<\mathrm{\mid }t\mathrm{\mid }\ll 1$. By interpreting the image of $f_t$ as the discriminant (the image of the critical point set) of a smooth map germ $H_t\text{\hspace{0.17em}⁣}:({\mathbb{C}}^5,0)\to ({\mathbb{C}}^4,0)$, we define an algebra whose dimension over $\mathbb{C}$ is equal to the number of $D_4$ points in the image of $f_t$.