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This chapter introduces two new geometric invariants that count the number of inflections and vertices concentrated at the singular point of the discriminant of a map-germ from the plane to the plane. The first invariant is preserved under the action of the product of the group of diffeomorphisms in the source and the affine group in the target. The second invariant is preserved under the product of the group of diffeomorphisms in the source and the group of similarities in the target. Properties of these invariants are investigated. As an application, their values are computed for the cusp, swallowtail, lips, beaks, butterfly, goose, and gull singularities.

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Geometric Invariants

  • 章节

  • 发表日期：

    2025年

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  • Farid Tari,
  • Mostafa Salarinoghabi,
  • Masaru Hasegawa

摘要

This chapter introduces two new geometric invariants that count the number of inflections and vertices concentrated at the singular point of the discriminant of a map-germ from the plane to the plane. The first invariant is preserved under the action of the product of the group of diffeomorphisms in the source and the affine group in the target. The second invariant is preserved under the product of the group of diffeomorphisms in the source and the group of similarities in the target. Properties of these invariants are investigated. As an application, their values are computed for the cusp, swallowtail, lips, beaks, butterfly, goose, and gull singularities.