Algebraic realization of stable Poincar\'e-Reeb graphs

Summary · qwen2.5:32b

The study establishes that any finite graph with a good orientation and vertices of degree 1 or 3 can be realized as the Poincaré-Reeb graph of a stable algebraic domain in $\mathbb{R}^n$ for $n \geq 2$, extending to graphs with vertices of degree 2 when $n \geq 3$. This work uses advanced algebraic approximation techniques, including recent extensions over $\mathbb{Q}$ by Ghiloni and the author, to achieve these realizations.

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Excerpt

arXiv:2602.18780v2 Announce Type: replace Abstract: 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 Poincar\'e-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 \`a 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.