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In this note, one already known result on the subject of homogeneous quadratic mappings is addressed. It concerns certain sufficient conditions for the existence of a regular zero in the framework of the so-called index approach. This result is revisited as some new proof is proposed which is shorter than the previous proof. At the same time, the invoked reasonings limit us to considering the case when the mapping image is in \(\mathbb {R}^3\) . Some further development in this direction is also discussed and, based on the proposed proof, a refinement of the main result is formulated. A few examples are analysed in this context.

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A Remark on the Existence of a Regular Zero of a Homogeneous Quadratic Mapping Into \(\mathbb {R}^3\)

  • 会议论文

  • 发表日期：

    2026年

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  • Dmitry Karamzin

摘要

In this note, one already known result on the subject of homogeneous quadratic mappings is addressed. It concerns certain sufficient conditions for the existence of a regular zero in the framework of the so-called index approach. This result is revisited as some new proof is proposed which is shorter than the previous proof. At the same time, the invoked reasonings limit us to considering the case when the mapping image is in \(\mathbb {R}^3\) . Some further development in this direction is also discussed and, based on the proposed proof, a refinement of the main result is formulated. A few examples are analysed in this context.