Algebraic realization of stable Poincar\'e-Reeb graphs

TL;DR

This paper demonstrates that a broad class of finite graphs can be realized as Poincaré-Reeb graphs of stable algebraic domains in Euclidean spaces, using advanced algebraic approximation techniques.

Contribution

It establishes a general realization theorem for graphs as Poincaré-Reeb graphs of algebraic domains, extending previous work and incorporating new algebraic approximation methods.

Findings

Graphs with vertices of degree 1 or 3 can be realized in any dimension.

In dimensions three and higher, graphs with vertices of degree 2 are also realizable.

Algebraic approximation techniques enable polynomial coefficient reduction and extension over real closed fields.

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.