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

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.

💡 Research Summary

The paper investigates planar regions whose boundaries are given by real algebraic curves of degree one or two—namely lines, circles, parabolas, and hyperbolas—and studies the combinatorial structure that arises when these regions are projected onto the x‑axis. By restricting the canonical projection π₂,₁ : ℝ² → ℝ (the first‑coordinate map) to a closed region D bounded by such curves, the authors consider the level sets of π₂,₁|_D. Each connected component of a level set is collapsed to a point, producing a quotient space that is a graph; this graph is called the “Poincaré‑Reeb graph” of the region. The construction is a natural extension of the classical Reeb graph, which is usually defined for smooth functions on manifolds, to the setting where the underlying space is a real algebraic region possibly with singular boundary components.

The central result (Main Theorem 0, restated as Main Theorem 1) asserts that every finite tree can be realized as the Poincaré‑Reeb graph of a region bounded by two families of parabolas. One family consists of upward‑opening parabolas of the form y = a(x – p₁)² + p₂ with a > 0, and the other consists of their downward‑opening counterparts y = –a(x – p₁)² + p₂. By arranging a suitable finite collection of such parabolas in the plane, the authors construct a region whose projection graph is isomorphic to any prescribed tree. A secondary result (Main Theorem 2) shows that the same statement holds when circles replace one of the families of parabolas.

The proof proceeds in two major stages. First, the authors treat trees whose vertices have degree different from two (i.e., trees without long linear chains). In this case a simple alternating arrangement of an upward and a downward parabola already yields a “linear” graph D₁ whose Poincaré‑Reeb graph is a single edge. By adding more parabolas in a cyclic fashion—so that each parabola intersects only its immediate neighbours—the authors obtain a basic graph homeomorphic to a star or a small tree.

Second, for a general tree (including vertices of degree two), the construction starts from the basic configuration above and inserts additional parabolas to create new vertices along existing edges. The insertion is performed locally: a point on an edge is chosen, a new parabola of sufficiently large curvature a₀ is placed very close to that point, and because its graph is almost vertical, the projection π₂,₁ splits the original edge into two (or more) edges with a new vertex at the intersection. The authors distinguish three insertion patterns: (i) adding one vertex at a point of the form (p₁,0,q), (ii) adding one vertex at a point (p₁,j,qₚ) lying on a previously placed parabola, and (iii) adding two vertices by inserting a parabola in a region that does not intersect any existing curve except at its endpoints. By iterating these operations arbitrarily many times, any finite tree can be built.

Technical ingredients include: (a) the use of the monotonicity of a parabola’s y‑coordinate to guarantee that each level set of π₂,₁ intersects the region in a single connected component; (b) careful control of intersection points so that they are isolated and correspond exactly to graph vertices; (c) the observation that the intersection of a vertical line {x = b} with the region is either a single point or a closed interval homeomorphic to D¹, which ensures that the quotient map is well‑behaved.

The paper also provides a thorough background on “RA‑regions” (real algebraic regions) and defines the Poincaré‑Reeb graph formally via the quotient of the region by the equivalence relation that identifies points lying in the same connected component of a level set. It reviews related literature, notably the recent work of Bodin, Popescu‑Pampu, and Sorea on embedding arbitrary finite graphs in the plane with disjoint real algebraic curves, and the author’s own series of preprints on explicit constructions of real algebraic maps (so‑called moment maps) from spheres to disks.

In the concluding discussion, the author emphasizes that this work bridges real algebraic geometry, Morse‑type theory, and combinatorial topology by showing that the combinatorial data of any tree can be encoded in the geometry of very simple algebraic curves. Limitations are acknowledged: the method is restricted to trees (acyclic graphs) and to degree‑two curves; extending the construction to graphs with cycles or to higher‑degree curves remains open. Moreover, while existence of sufficiently large curvature parameters (a₀) is proved abstractly, explicit quantitative bounds are not provided, which would be necessary for computational implementations.

Overall, the paper contributes a concrete and constructive realization theorem for trees as Poincaré‑Reeb graphs of regions bounded by parabolas, enriching the toolbox for studying the interplay between real algebraic sets and their associated topological invariants.