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Regions surrounded by parabolas in the plane and trees representing their shapes respecting their natural projection to the line

Naoki Kitazawa

TL;DR

The work addresses realizing regions enclosed by real algebraic curves of degree one or two in the plane by their combinatorial Poincaré-Reeb graphs. It presents a constructive method showing that every finite tree can be realized as the PR graph of a RA-region bounded by parabolas of two congruence types, starting from a base two-parabola configuration and inserting additional parabolas to create new vertices along edges, with a supplementary step to handle degree-two vertices. The approach connects real algebraic geometry with singularity theory via moment-map-like constructions and builds on recent results by Bodin, Popescu-Pampu, and Sorea. The results provide explicit geometric realizations and contribute to the explicit realization of graphs as Reeb-type invariants, with potential relevance to explicit algebraic maps onto regions and their moment-map interpretations.

Abstract

The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics being also elementary and connected to various fields. The shapes are understood via graphs the regions collapsing to respecting the canonical projection onto the 1st component. Our main result is the following: each tree is realized by regions surrounded by parabolas of two types, here. Related studies are elementary and interesting and surprisingly, this explicit field is started very recently, by Bodin, Popescu-Pampu and Sorea in the 2020s. After that, this is developing, due to the author. The author also investigates this motivated by studies on explicit construction of real algebraic maps onto the regions locally so-called moment maps: this comes from singularity theory of differentiable maps and real algebraic geometry.

Regions surrounded by parabolas in the plane and trees representing their shapes respecting their natural projection to the line

TL;DR

The work addresses realizing regions enclosed by real algebraic curves of degree one or two in the plane by their combinatorial Poincaré-Reeb graphs. It presents a constructive method showing that every finite tree can be realized as the PR graph of a RA-region bounded by parabolas of two congruence types, starting from a base two-parabola configuration and inserting additional parabolas to create new vertices along edges, with a supplementary step to handle degree-two vertices. The approach connects real algebraic geometry with singularity theory via moment-map-like constructions and builds on recent results by Bodin, Popescu-Pampu, and Sorea. The results provide explicit geometric realizations and contribute to the explicit realization of graphs as Reeb-type invariants, with potential relevance to explicit algebraic maps onto regions and their moment-map interpretations.

Abstract

The author has been interested in regions surrounded by real algebraic curves of degree or in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics being also elementary and connected to various fields. The shapes are understood via graphs the regions collapsing to respecting the canonical projection onto the 1st component. Our main result is the following: each tree is realized by regions surrounded by parabolas of two types, here. Related studies are elementary and interesting and surprisingly, this explicit field is started very recently, by Bodin, Popescu-Pampu and Sorea in the 2020s. After that, this is developing, due to the author. The author also investigates this motivated by studies on explicit construction of real algebraic maps onto the regions locally so-called moment maps: this comes from singularity theory of differentiable maps and real algebraic geometry.
Paper Structure (5 sections, 1 theorem, 1 equation, 1 figure)

This paper contains 5 sections, 1 theorem, 1 equation, 1 figure.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Reviewing RA-regions and their Poincaré-Reeb graphs.
  4. Main Theorem \ref{['mthm:0']} revisited as Main Theorem \ref{['mthm:1']}, and Main Theorem \ref{['mthm:2']}.
  5. Conflict of interest and Data availability.

Key Result

Theorem 1

For circles instead of parabolas, the same fact holds.

Figures (1)

  • Figure 1: $(D,\{S_j\}_{j=1}^l)$ for STEP 1-1-2. Its Poincaré-Reeb graph is colored in grey. Dotted straight segments show segments parallel to the normal vectors at the points. Hereafter, each "$\cdots$" in Figures is for abbreviation of certain objects.

Theorems & Definitions (3)

  • proof
  • Theorem 1
  • proof