On the non-existence of certain real algebraic surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912629951299584 |
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| author | Guadarrama-García, Miguel Angel |
| author_facet | Guadarrama-García, Miguel Angel |
| contents | In this note is given an algebraic solution to the problem 1997-6 proposed by D. A. Panov in the list of Arnold's problems \cite{Arnld2b}. In particular, it is shown that there does not exist a real polynomial function $f$ on the real euclidean plane, whose Hessian is positive in an open set bordered by smooth connected curve, and the parabolic curve of the graph of $f$ has only one special parabolic point with index $+1$. Besides, we find conditions on $f$ so that its graph has more special parabolic points with index -1 than with index +1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the non-existence of certain real algebraic surfaces Guadarrama-García, Miguel Angel Differential Geometry 53A15, 53A05, 14PO5, 14N10, 34K32, 34G20 In this note is given an algebraic solution to the problem 1997-6 proposed by D. A. Panov in the list of Arnold's problems \cite{Arnld2b}. In particular, it is shown that there does not exist a real polynomial function $f$ on the real euclidean plane, whose Hessian is positive in an open set bordered by smooth connected curve, and the parabolic curve of the graph of $f$ has only one special parabolic point with index $+1$. Besides, we find conditions on $f$ so that its graph has more special parabolic points with index -1 than with index +1. |
| title | On the non-existence of certain real algebraic surfaces |
| topic | Differential Geometry 53A15, 53A05, 14PO5, 14N10, 34K32, 34G20 |
| url | https://arxiv.org/abs/2510.04025 |