Invariants of wavefronts (wfinvariants)

Let G\colon (\mathbb{R}^n,0) \to (\mathbb{R}^n,0) be a smooth, corank 1 map with weighted homogeneous components and isolated singularity. If we parametrise G as (x,y) \mapsto (x,g(x,y)) with x \in \mathbb{R}^{n-1} and y \in \mathbb{R}, [1] gives formulas to count the zero-schemes in the discriminant of G. In particular, a result by Arnol'd states that \Delta(G) is a front if and only if g_y is a regular function, so this gives a method to count the zero-schemes on a front.

If g has weights (w_1,w_2,w_3) and degree d,

S=\frac{(d-w_3)(d-2w_3)(d-3w_3)}{w_1w_2w_3};

K=S(d-4w_3);

T=\frac{1}{6}K(d-5w_3)