Algebraic realization of stable Poincaré-Reeb graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912917473984512 |
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| author | Savi, Enrico |
| author_facet | Savi, Enrico |
| contents | We introduce the notion of domain of finite type $\mathscr{D}\subset\mathbb{R}^n$ generalizing an earlier work of Bodin, Popescu-Pampu and Sorea. Then, we prove that every finite graph admitting a good orientation whose vertices have degree 1 or 3 can be realized as the Poincaré-Reeb graph of a stable (globally) algebraic domain of finite type $\mathscr{D}\subset\mathbb{R}^n$, for every $n\geq 2$. If in addition $n\geq 3$, we construct a class of graphs allowing vertices of degree $2$ also. Algebraic approximation techniques à la Nash-Tognoli and stable Morse functions are fundamental tools in our approach. In particular, the recent extensions over $\mathbb{Q}$ of such algebraic approximation techniques developed by Ghiloni and the author allow us to reduce the coefficients of the describing polynomials over $\mathbb{Q}$ and to extend our constructions over real closed fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18780 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic realization of stable Poincaré-Reeb graphs Savi, Enrico Algebraic Geometry Primary 14P05, 14P25, Secondary 58K05, 14P20 We introduce the notion of domain of finite type $\mathscr{D}\subset\mathbb{R}^n$ generalizing an earlier work of Bodin, Popescu-Pampu and Sorea. Then, we prove that every finite graph admitting a good orientation whose vertices have degree 1 or 3 can be realized as the Poincaré-Reeb graph of a stable (globally) algebraic domain of finite type $\mathscr{D}\subset\mathbb{R}^n$, for every $n\geq 2$. If in addition $n\geq 3$, we construct a class of graphs allowing vertices of degree $2$ also. Algebraic approximation techniques à la Nash-Tognoli and stable Morse functions are fundamental tools in our approach. In particular, the recent extensions over $\mathbb{Q}$ of such algebraic approximation techniques developed by Ghiloni and the author allow us to reduce the coefficients of the describing polynomials over $\mathbb{Q}$ and to extend our constructions over real closed fields. |
| title | Algebraic realization of stable Poincaré-Reeb graphs |
| topic | Algebraic Geometry Primary 14P05, 14P25, Secondary 58K05, 14P20 |
| url | https://arxiv.org/abs/2602.18780 |