Initial Newton polynomial of the discriminant

Abstract

Let (𝑓, g)∶ (ℂ 2 , 0) ⟶ (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 rescaling variables, by the ideals (𝑓) and (g).M S C ( 2 0 2 0 ) 32S15 (primary), 32S45 (secondary) INTRODUCTIONLet ℝ ⩾0 (ℤ ⩾0 ) be the set of all nonnegative real (integer) numbers. For a power series 𝑓 = ∑ (𝑖,𝑗)∈ℤ 2 ⩾0 𝑎 𝑖,𝑗 𝑥 𝑖 𝑦 𝑗 ∈ ℂ[[𝑥, 𝑦]] we define its Newton diagram Δ(𝑓) as the convex hull of the union ⋃ 𝑎 𝑖,𝑗 ≠0 ((𝑖, 𝑗) + ℝ 2 ⩾0). If 𝑆 is the union of all compact edges of Δ(𝑓), then the polynomial 𝑓| 𝑆 ∶= ∑ (𝑖,𝑗)∈𝑆 𝑎 𝑖,𝑗 𝑥 𝑖 𝑦 𝑗 is called the initial Newton polynomial of 𝑓. We say that power series 𝑓 1 , 𝑓 2 ∈ ℂ[ [𝑥, 𝑦]] are equal up to rescaling variables if 𝑓 2 (𝑥, 𝑦) = 𝑓 1 (𝑎𝑥, 𝑏𝑦) for some nonzero constants 𝑎, 𝑏.Let 𝜙 = (𝑓, g)∶ (ℂ 2 , 0) → (ℂ 2 , 0) be the germ of a holomorphic mapping with an isolated zero at the origin. To any germ 𝜉 of an analytic curve in (ℂ 2 , 0) one associates its direct image 𝜙 * (𝜉), see, for example, [3,4]. The direct image of 𝜉 by 𝜙 is an analytic curve germ in the target space uniquely determined by the following two properties.(i) If 𝜉 ⊂ (ℂ 2 , 0) is an irreducible curve then 𝜙 * (𝜉) is the curve of equation 𝐻 𝑑 = 0, where 𝐻 = 0 is a reduced equation of the curve 𝜙(𝜉) in the target space and 𝑑 is the topological degree of the restriction 𝜙| 𝜉 ∶ 𝜉 → 𝜙(𝜉). (ii) If ℎ = ℎ 1 ⋯ ℎ 𝑠 is a factorization of a power series ℎ to the product of irreducible factors in ℂ{𝑥, 𝑦}, then 𝜙 * ({ℎ = 0}) is the curve 𝐻 1 ⋯ 𝐻 𝑠 = 0, where the curves 𝐻 𝑖 = 0 are the direct images of the branches ℎ 𝑖 = 0 for 𝑖 = 1, … , 𝑠.The direct image can be also characterized as follows. The direct image of a curve germ ℎ = 0 is the only curve germ 𝐻 = 0 that satisfies the projection formula: For any analytic curve 𝑤 = 0 in the target space we have the equality of intersection multiplicities 𝑖 0 (𝑤•𝜙, ℎ) = 𝑖 0 (𝑤, 𝐻).