Finite A-determinacy of generic homogeneous map germs in ℂ³

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  • Finite 𝒜-determinacy of generic homogeneous map germs in ℂ<sup>3</sup>
  • Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$

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<p>Denote by 𝐻(𝑑1, 𝑑2, 𝑑3) the set of all homogeneous polynomial mappings 𝐹 = (𝑓1, 𝑓2, 𝑓3) : ℂ3 → ℂ3, such that deg 𝑓𝑖 = 𝑑𝑖. We show that if gcd(𝑑𝑖, 𝑑𝑗) ≤ 2 for 1 ≤ 𝑖 < 𝑗 ≤ 3 and gcd(𝑑1, 𝑑2, 𝑑3) = 1, then there is a non-empty Zariski open subset 𝑈 ⊂ 𝐻(𝑑1, 𝑑2, 𝑑3) such that for every mapping 𝐹 ∈ 𝑈 the map germ (𝐹, 0) is 𝒜-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping (𝑓1, 𝑓2, 𝑓3): ℂ3 → ℂ3, where deg 𝑓𝑖 = 𝑑𝑖.</p>

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