2014/03/21 by Justyna Bobowik, Bobowik, Justyna, Zbigniew Szafraniec +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.AG
paper · pdf · doi:10.48550/arxiv.1403.5379
arxiv created 2014/03/21 · openalex publication_date 2014/03/21 · arxiv updated 2014/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be an oriented 3-manifold. For a generic f ∈ C^ ∞(M,R3), there is a discrete set of swallowtail critical points. In that case, at any swallowtail point p there exists a well-oriented coordinate system centered at p, and a coordinate system centered at f(p), such that locally f has the form f_±(x,y,z)=(± xy+x2 z+x4,y,z), so one may associate with p a sign I(f,p)∈ \± 1\. A geometric definition of the sign associated with a swallowtail was recently introduced by Goryunov. We shall show how to compute the number of swallowtail points having the positive/negative sign, in the case where f : Rn → Rn is a polynomial mapping, in terms of signatures of quadratic forms.