Refined algebraic domains with finite sets in the boundaries

TL;DR

This work extends the theory of refined algebraic domains by introducing poles, i.e., finite boundary data organizing how new algebraic curves are added to existing domains. By defining LS/PS/PLS and the pole triplet $(\mathcal S,D_{\mathcal S},A_{D_{\mathcal S}})$, the author systematizes how adding disks, circles and ellipsoids affects the associated Poincar-Reeb graphs, viewed as V-digraphs, and proves stability results under controlled deformations. In the circle case, precise geometric and combinatorial outcomes are established: the intersection of added disks with the boundary region yields at most a single boundary point in the pole set, and the Poincar-Reeb graphs persist under isotopies, enabling explicit constructions that realize given graph structures via real algebraic maps. The work connects to Reeb-graph reconstruction, singularity theory, and explicit polynomial approximations, offering a framework to design refined domains whose projection graphs realize prescribed combinatorial types with controlled local interactions. Overall, the paper broadens the toolbox for shaping algebraic domains and their boundary-induced graphs, with potential applications in real algebraic geometry and singularity theory.

Abstract

Refined algebraic domains are regions in the plane surrounded by finitely many non-singular real algebraic curves which may intersect with normal crossing. We are interested in shapes of such regions with surrounding real algebraic curves. Poincar'e-Reeb Graphs of them are graphs the regions naturally collapse to respecting the projection to a straight line. Such graphs were first formulated by Sorea, for example, around 2020, and regions surrounded by mutually disjoint non-singular real algebraic curves were mainly considered. The author has generalized the studies to several general situations. We find classes of such objects defined inductively by adding curves. We respect characteristic finite sets in the curves. We consider regions surrounded by the curves and of a new type. We investigate geometric properties and combinatorial ones of them and discuss important examples. We also previously studied explicit classes defined inductively in this way and review them.

Figures (5)

• Figure 1: A region surrounded by two concentric circles whose radii are sufficiently close, a closed disk $D_{j^{\prime}}$, colored in blue, and the black colored boundary of the closed disk $\overline{D_{j^{\prime}}}$. Black dots are for points of $F_{D_{\mathcal{S}},1} \bigcup F_{D_{\mathcal{S}},2}$.
• Figure 2: The family $\mathcal{S}$ of circles consists of exactly three circles. Two of the circles are of a same radius and centered at two points in a line parallel to the horizontal line $\mathbb{R} \times \{0\}$. The disk $D_{j^{\prime}}$ with a point $x_{j^{\prime}} \in D_{j^{\prime}}$ contained in the two circles from $\mathcal{S}$ is added. The region $D_{\mathcal{S}}$ ($D_{\mathcal{S}} \bigcap ({\mathbb{R}}^2-\overline{D_{j^{\prime}}})$) is colored in gray and the disk $D_{j^{\prime}}$ is colored in blue. The dotted line is parallel to the vertical line $\{0\} \times \mathbb{R} \subset {\mathbb{R}}^2$.
• Figure 3: A region surrounded by two concentric circles and an ellipsoid $D_{j^{\prime}}$, colored in blue, and having the black colored boundary. This is for (the additional statement of) Theorem \ref{['thm:1']}.
• Figure 4: A pair of ellipsoids of the standard form containing the two points $({\pi}_{2,1,1}(v_0)-{\epsilon}_{v_0,+,j},a_{v_0,+,j})$ and $({\pi}_{2,1,1}(v_0)-{\epsilon}_{v_0,+,j+1},a_{v_0,+,j+1})$, colored in black with blue colored contours.
• Figure 5: The desired ellipsoid $D_{j^{\prime}}$ of the standard form centered at a point of the form $(\min \{{\pi}_{2,1,1}(v_0)-{\epsilon}_{v_0,+,j},{\pi}_{2,1,1}(v_0)-{\epsilon}_{v_0,+,j+1}\}(={\pi}_{2,1,1}(v_0)-{\epsilon}_{v_0,+,j},a_{v_0,+,j,j+1})$. This respects FIGURE \ref{['fig:4']}.

Theorems & Definitions (14)

• Corollary 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4: kitazawa7
• proof : Reviewing the original proof
• Proposition 1