Finite ๐’œ-determinacy of generic homogeneous map germs in โ„‚<sup>3</sup>

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  • Finite A-determinacy of generic homogeneous map germs in โ„‚ยณ
  • Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$

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<p>Denote by ๐ป(๐‘‘1, ๐‘‘2, ๐‘‘3) the set of all homogeneous polynomial mappings ๐น = (๐‘“1, ๐‘“2, ๐‘“3) : โ„‚3 → โ„‚3, such that deg ๐‘“๐‘– = ๐‘‘๐‘–. We show that if gcd(๐‘‘๐‘–, ๐‘‘๐‘—) ≤ 2 for 1 ≤ ๐‘– < ๐‘— ≤ 3 and gcd(๐‘‘1, ๐‘‘2, ๐‘‘3) = 1, then there is a non-empty Zariski open subset ๐‘ˆ ⊂ ๐ป(๐‘‘1, ๐‘‘2, ๐‘‘3) such that for every mapping ๐น ∈ ๐‘ˆ the map germ (๐น, 0) is ๐’œ-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping (๐‘“1, ๐‘“2, ๐‘“3): โ„‚3 → โ„‚3, where deg ๐‘“๐‘– = ๐‘‘๐‘–.</p>

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