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January, 2021 Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$
Michał FARNIK, Zbigniew JELONEK, Maria Aparecida Soares RUAS
J. Math. Soc. Japan 73(1): 211-220 (January, 2021). DOI: 10.2969/jmsj/83208320

Abstract

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \mathbb{C}^3 \to \mathbb{C}^3$, such that $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j) \leq 2$ for $1 \leq i < j\leq 3$ and $\gcd(d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U \subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$ is $\mathcal{A}$-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping $(f_1,f_2,f_3):\mathbb{C}^3 \to \mathbb{C}^3$, where $\deg f_i=d_i$.

Funding Statement

The authors are partially supported by the grant of Narodowe Centrum Nauki, grant number 2015/17/B/ST1/02637, additionally the third author is partially supported by the FAPESP grant 2014/00304-2.

Citation

Download Citation

Michał FARNIK. Zbigniew JELONEK. Maria Aparecida Soares RUAS. "Finite $\mathcal{A}$-determinacy of generic homogeneous map germs in $\mathbb{C}^3$." J. Math. Soc. Japan 73 (1) 211 - 220, January, 2021. https://doi.org/10.2969/jmsj/83208320

Information

Received: 28 August 2019; Revised: 16 September 2019; Published: January, 2021
First available in Project Euclid: 10 June 2020

Digital Object Identifier: 10.2969/jmsj/83208320

Subjects:
Primary: 14R99
Secondary: 32A10‎

Rights: Copyright © 2021 Mathematical Society of Japan

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Vol.73 • No. 1 • January, 2021
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