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Arrangements of circles supported by small chords and compatible with natural real algebraic functions

Naoki Kitazawa

TL;DR

The paper introduces a novel class of circle arrangements in the plane, SSC-NI, within the broader NI framework and connects these geometric configurations to real algebraic maps that yield Morse-Bott–type functions via projections. It develops a formal, graph-theoretic description of the surrounding regions through Poincaré-Reeb V-digraphs and relates these to Reeb graphs of associated functions using Ehresmann-type fibrations. A key contribution is the explicit analysis of local changes in the Poincaré-Reeb graphs when adding small chords and circles, providing constructive mechanisms to realize regions as images of real algebraic maps. The results broaden explicit constructions in real algebraic geometry and singularity theory, enabling explicit realizations of regions bounded by circle arrangements and their global structure understanding. Practically, the work offers a framework to generate and manipulate Morse-Bott–type maps and their associated combinatorial graphs from circle configurations, with explicit examples and potential applications in explicit real algebraic geometry.

Abstract

We have previously proposed a study of arrangements of small circles which also surround regions in the plane realized as the images of natural real algebraic maps yielding Morse-Bott functions by projections. Among studies of arrangements, families of smooth regular submanifolds in smooth manifolds, this study is fundamental, explicit, and new, surprisingly. We have obtained a complete list of local changes of the graphs the regions naturally collapse to in adding a (generic) small circle to an existing arrangement of the proposed class. Here, we propose a similar and essentially different class of arrangements of circles. The present study also yields real algebraic maps and nice real algebraic functions similarly and we present a similar study. We are interested in topological properties and combinatorics among such arrangements and regions and applications to constructing such real algebraic maps and manifolds explicitly and understanding their global structures.

Arrangements of circles supported by small chords and compatible with natural real algebraic functions

TL;DR

The paper introduces a novel class of circle arrangements in the plane, SSC-NI, within the broader NI framework and connects these geometric configurations to real algebraic maps that yield Morse-Bott–type functions via projections. It develops a formal, graph-theoretic description of the surrounding regions through Poincaré-Reeb V-digraphs and relates these to Reeb graphs of associated functions using Ehresmann-type fibrations. A key contribution is the explicit analysis of local changes in the Poincaré-Reeb graphs when adding small chords and circles, providing constructive mechanisms to realize regions as images of real algebraic maps. The results broaden explicit constructions in real algebraic geometry and singularity theory, enabling explicit realizations of regions bounded by circle arrangements and their global structure understanding. Practically, the work offers a framework to generate and manipulate Morse-Bott–type maps and their associated combinatorial graphs from circle configurations, with explicit examples and potential applications in explicit real algebraic geometry.

Abstract

We have previously proposed a study of arrangements of small circles which also surround regions in the plane realized as the images of natural real algebraic maps yielding Morse-Bott functions by projections. Among studies of arrangements, families of smooth regular submanifolds in smooth manifolds, this study is fundamental, explicit, and new, surprisingly. We have obtained a complete list of local changes of the graphs the regions naturally collapse to in adding a (generic) small circle to an existing arrangement of the proposed class. Here, we propose a similar and essentially different class of arrangements of circles. The present study also yields real algebraic maps and nice real algebraic functions similarly and we present a similar study. We are interested in topological properties and combinatorics among such arrangements and regions and applications to constructing such real algebraic maps and manifolds explicitly and understanding their global structures.
Paper Structure (10 sections, 9 theorems, 7 figures)

This paper contains 10 sections, 9 theorems, 7 figures.

Table of Contents

  1. Introduction.
  2. Notation on topological spaces, manifolds and maps.
  3. Our arrangements of circles.
  4. Organization of our paper and our main work.
  5. Poincaré-Reeb graphs of the region $D_{\mathcal{S}}$ and real algebraic maps onto (the closure) of the region.
  6. Dimensions of topological spaces decomposed into cells, smooth maps and graphs.
  7. A Poincaré-Reeb graph, our region surrounded by the circles collapses.
  8. A real algebraic map onto the closure $\overline{D_{\mathcal{S}}} \subset {\mathbb{R}}^2$.
  9. Our new class of arrangements of circles: SSC-NI arrangements.
  10. Conflict of interest and Data availability.

Key Result

Proposition 1

For an arbitrary chord of $\mathcal{S}$ obtained above, either it is always a subset of $\overline{D_{\mathcal{S}}} \bigcap {D^{\prime}}_{x_{j^{\prime},0}}$ or a subset of $({\mathbb{R}}^2-D_{\mathcal{S}}) \bigcap {D^{\prime}}_{x_{j^{\prime},0}}$.

Figures (7)

  • Figure 1: Chords of $\mathcal{S}$ of three types for Proposition \ref{['prop:2']} (\ref{['prop:2.3']}).
  • Figure 2: A supported CS-region at $(D_{\mathcal{S}},x_{j^{\prime},0})$ for Theorem \ref{['thm:1']} (\ref{['thm:1.1']}) and Theorem \ref{['thm:2']} (\ref{['thm:2.1']}).
  • Figure 3: An unsupported CS-region at $(D_{\mathcal{S}},x_{j^{\prime},0})$ for Theorem \ref{['thm:1']} (\ref{['thm:1.1']}) and Theorem \ref{['thm:2']} (\ref{['thm:2.2.2']}).
  • Figure 4: The intersection of a supported CS-region at $(D_{\mathcal{S}},x_{j^{\prime},0})$ and $\overline{D_{\mathcal{S}}}$ for Theorem \ref{['prop:2']} (\ref{['prop:2.2']}) and Theorem \ref{['thm:5']}.
  • Figure 5: Four types of chords at $(D_{\mathcal{S}},x_{j^{\prime},0})$ for Theorem \ref{['thm:3']} (\ref{['thm:3.1.1']}).
  • ...and 2 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof : Notes on our proof of Theorem \ref{['thm:1']}
  • Theorem 2
  • proof : Notes on our proof of Theorem \ref{['thm:2']}
  • Proposition 3
  • Theorem 3
  • proof : Notes on our proof of Theorem \ref{['thm:3']}
  • Theorem 4
  • ...and 4 more