Initial Newton polynomial of the discriminant

Let $(f,g)\text{\hspace{0.17em}⁣}:({\mathbb{C}}^2,0)⟶({\mathbb{C}}^2,0)$ be a holomorphic mapping with an isolated zero. We show that the initial Newton polynomial of its discriminant is determined, up to rescalling variables, by the ideals $(f)$ and $(g)$.