Algebraic realization of stable Poincar\'e-Reeb graphs

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.