Non-degeneracy of the discriminant

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).

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“…This includes a characterization of the atypical values of the pencils

f^k=tg^l

and of the asymptotic critical values of the meromorphic functions

f^k/g^l

. We also propose an alternative proof of [6, Theorem 6.6] by an argument similar to the proof of the main result.…”

Section: Introductionmentioning

confidence: 92%

Initial Newton polynomial of the discriminant

Gryszka

Gwoździewicz

Parusiński

2022

Bulletin of London Math Soc

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show abstract

“…This includes a characterization of the atypical values of the pencils f k = tg l and of the asymptotic critical values of the meromorphic functions f k /g l . We also propose an alternative proof of [GGL,Theorem 6.6] by an argument similar to the proof of the main result.…”

Section: Introductionmentioning

confidence: 93%

“…Theorem 5.4 ( [GGL,Theorem 6.6]) Let h = 0 be a unitangent singular curve and let ℓ 1 = 0, ℓ 2 = 0 be smooth curves transverse to h = 0. Then there exists a nonzero constant d ∈ C such that the initial Newton polynomials of discriminants of mappings (dℓ 1 , h) : (C 2 , 0) → (C 2 , 0) and (ℓ 2 , h) : (C 2 , 0) → (C 2 , 0) are equal.…”

mentioning

confidence: 99%

Initial Newton polynomial of the discriminant

Gryszka,

Gwoździewicz,

Parusiński

2021

Preprint

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“…Proof For genus 2, and after the second part of [5,Corollary 4.4] we get that D(u, v) = 0 is nondegenerate and we can determine its topological type from its Newton polygon (see [13,Proposition 4.7] and [6, Theorem 3.2]), which is the jacobian Newton polygon of (x, f ) (see Fig. 1).…”

Section: Proposition 33mentioning

confidence: 99%

“…In [5] the authors study the pairs ( , f ) for which the discriminant curve is nondegenerate in the Kouchnirenko sense. In particular, when f is irreducible, after [5,Corollary 4.4] the discriminant curve D(u, v) = 0 is nondegenerate if and only if the multiplicity of f (x, y) = 0 equals 2 or equals 4 and the genus equals 2. Otherwise the discriminant curve is degenerate.…”

Section: Introductionmentioning

confidence: 99%

Topological type of discriminants of some special families

Barroso

Iglesias

2021

Period Math Hung

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We will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

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Non-degeneracy of the discriminant

Cited by 5 publications

References 17 publications

Initial Newton polynomial of the discriminant

Initial Newton polynomial of the discriminant

Initial Newton polynomial of the discriminant

Topological type of discriminants of some special families

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