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**Elliptic domains of general families of homogeneous polynomials and extreme functions.**
*(English.
Russian original)*
Zbl 0584.58008

Funct. Anal. Appl. 19, 102-111 (1985); translation from Funkts. Anal. Prilozh. 19, No. 2, 26-36 (1985).

The author investigates singularities of the boundary of the elliptic domains for general families of homogeneous polynomials. To be precise, let \(\Lambda\) be an \(\ell\)-dimensional smooth manifold, \(P^ d_ n\) the space of all homogeneous real polynomials of degree d in n variables and a smooth map \(\phi\) : \(\Lambda\) \(\to P^ d_ n\), called a family of homogeneous polynomials of degree d with base \(\Lambda\). The elliptic domain el \(\phi\) is the set of all \(\lambda\in \Lambda\), for which \(\phi\) (\(\lambda)\) is elliptic. The investigation is based on the fact that the singularities of the boundary \(\partial (el \phi)\) can be identified with the singularities of level sets of an extreme function \(\min_{x\in S^{n-1}}\phi (\cdot)(x)\); \(\max_{x\in S^{n-1}}\phi (\cdot)(x)\) \((\in C^{\infty}(\Lambda,{\mathbb{R}}))\) so the author can prove that the domain of ellipticity and its boundary in the neighborhood of a boundary point are diffeomorphic to the subgraph and graph of the maximum function of a suitable family in general position. Furthermore, he classifies the singularities of the boundaries of elliptic domains of families in general position for \(1\leq \ell \leq 7\), and shows that they stabilize for fixed \(\ell \in {\mathbb{N}}\) with increasing d and n. (i.e. they are identical for families of polynomials of all sufficiently large (even) degrees in a sufficiently large number of arguments).

Reviewer: B.Schomburg

### References:

[1] | V. I. Arnol’d et al., Singularities of Differentiable Maps. Classification of Singular Points, Caustics, and Wave Fronts [in Russian], Nauka, Moscow (1982). |

[2] | T. Brecker and D. Lander, Differentiable Germs and Catastrophes [Russian translation], Mir, Moscow (1977). |

[3] | L. N. Bryzgalova, ”Maximum functions of families of functions, depending on parameters,” Funkts. Anal. Prilozhen.,12, No. 1, 66-67 (1978). · Zbl 0405.58024 |

[4] | M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities [Russian translation], Mir, Moscow (1977). · Zbl 0429.58004 |

[5] | E. E. Landis, ”Tangential singularities,” Funkts. Analiz Prilozhen.,15, No. 2, 36-49 (1981). · Zbl 0493.58005 |

[6] | V. I. Matov, ”Topological classification of germs of maximum and minimum functions of families of functions in general position,” Usp. Mat. Nauk,37, No. 4, 167-168 (1982). · Zbl 0506.58004 |

[7] | A. T. Fomenko, Differential Geometry and Topology. Supplementary Chapters [in Russian], Moscow State Univ. (1983). · Zbl 0517.53001 |

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