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Algebraic realization of stable Poincaré-Reeb graphs

Enrico Savi

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 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.

Algebraic realization of stable Poincaré-Reeb graphs

Abstract

We introduce the notion of domain of finite type 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 , for every . If in addition , we construct a class of graphs allowing vertices of degree also. Algebraic approximation techniques à la Nash-Tognoli and stable Morse functions are fundamental tools in our approach. In particular, the recent extensions over of such algebraic approximation techniques developed by Ghiloni and the author allow us to reduce the coefficients of the describing polynomials over and to extend our constructions over real closed fields.
Paper Structure (8 sections, 8 theorems, 9 equations, 2 figures)

This paper contains 8 sections, 8 theorems, 9 equations, 2 figures.

Table of Contents

  1. Poincaré-Reeb graphs of domains of finite type in ${\mathbb R}^n$
  2. Domains of finite type in ${\mathbb R}^n$ and their Poincaré-Reeb graphs
  3. Topological constraints in realizing Poincaré-Reeb graphs
  4. Stable domains of finite type
  5. Algebraic and globally algebraic domains of finite type
  6. Algebraic realization of Poincaré-Reeb graphs
  7. Algebraic approximation of stable domains of finite type
  8. Algebraic realization of domains of finite type with given Poincaré-Reeb graph

Key Result

Proposition 1.5

Let $\mathscr{D}\subset{\mathbb R}^n$ be a domain of finite type. If for every $y\in{\mathbb R}$ the $k$-th fundamental group of each connected component of $\pi^{-1}(y)$ is trivial for every $k\leq n-1$, then $\mathscr{D}$ is homotopically equivalent to its Poincaré-Reeb graph $\mathcal{R}(\mathscr

Figures (2)

  • Figure 1.1: Example of a domain $\mathscr{D}\subset{\mathbb R}^n$ which does not retract to its Poincaré-Reeb graph $\mathcal{R}(\mathscr{D})\subset{\mathbb R}^n$ for $n\geq 3$.
  • Figure 1.2: Example of a graph $\Gamma$ which does not admit any good orientation.

Theorems & Definitions (39)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • proof
  • Remark 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 1.9
  • ...and 29 more