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

TL;DR

This paper explores the shapes of regions bounded by degree 1 or 2 algebraic curves, specifically parabolas, and demonstrates that any tree structure can be realized by such regions, advancing the understanding of their combinatorics and geometry.

Contribution

The paper proves that every tree can be represented by regions bounded by two types of parabolas, providing explicit constructions in the context of real algebraic geometry.

Findings

Any tree can be realized by regions bounded by parabolas.

The study connects algebraic curves with combinatorial tree structures.

It advances the explicit construction of real algebraic maps related to moment maps.

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.