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Reconstruction of real algebraic functions into curves with prescribed Reeb graphs

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Pith's one-line read The paper proves that any finite graph with vertices of degree 1 or 3 and a generic embedding into a curve is the Reeb graph of a real algebraic Morse function on a closed real algebraic manifold, for any dimension at least 2, and…

desk verdict Plausible new construction for curve-valued Reeb graphs, but Theorem 1's proof skips the step that makes it work; the explicit circle examples are the solid part. read the letter →

arxiv 2507.09467 v2 pith:5H44VCDT submitted 2025-07-13 math.AG math.COmath.GTmath.MG

Naoki Kitazawa This is my paper · ORCID
classification math.AGmath.COmath.GTmath.MG MSC 14P0514P1014P2014P2557R4558C05
keywords realalgebraicmanifoldsmapsReebgraphsMorsefunctionsunit-sphereconstructioncurve-valuedspecialgenericpolynomialapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to turn a purely combinatorial prescription into an explicit algebraic object: given a finite graph whose vertices have degree 1 or 3 and which embeds piecewise smoothly into a curve in a generic way, it claims there is a closed real algebraic manifold in Euclidean space and a real algebraic map to a neighborhood of the curve whose Reeb graph is exactly that graph. The Reeb graph records how the connected components of the level sets of a function merge and split; the simplest example is the height function on a sphere, whose Reeb graph has two vertices and one edge. The composed function to the curve is Morse, meaning its critical points are nondegenerate. The result matters because it shows these shapes are not just realizable by smooth functions but by functions defined by polynomial equations, so the whole construction is explicit and comes with degrees and manifold descriptions. This is the first treatment of the curve-valued case, extending earlier work where the target was the real line.

What carries the argument

The unit-sphere construction (US construction): given a compact connected semi-algebraic set $\{x \in \mathbb{R}^k \mid F(x) \geq 0\}$ surrounded by the regular zero set $\{F = 0\}$, form the hypersurface $\{(x,y) \in \mathbb{R}^{k+k'} \mid F(x) - \|y\|^2 = 0\}$, which is a regular real algebraic manifold; the canonical projection to $\mathbb{R}^k$ is a special generic map, locally of the form $(x_1,\dots,x_{n-1},\sum x_j^2)$, whose Reeb graph mirrors the boundary geometry. The construction reduces the graph-realization problem to approximating, in the Whitney topology, a regular neighborhood of the embedded graph by the zero set of a real polynomial, so that the projection to $C$ is Morse with exactly the prescribed Reeb graph.

What would settle it

Run the construction on a concrete graph and check the output: for instance, take a $\theta$ graph (two degree-3 vertices joined by three edges) embedded in a collar of a circle, approximate a regular neighborhood by a real polynomial $F$, form the hypersurface $\{(x,y) : F(x) - \|y\|^2 = 0\}$, and compute the Reeb graph of the projection to the circle. If the Reeb graph is not isomorphic to the original graph, or if the projection is not Morse, the central claim fails. Equivalently, find a graph satisfying the hypotheses for which the asserted Whitney approximation of the regular neighborhood cannot be made without creating extra vertices or edges.

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Extended reading notes

Core claim

Theorem 1 states: let $C$ be a one-dimensional connected regular real algebraic manifold in $\mathbb{R}^2$ with a product collar $NInt(C)$, and let $c_G \colon G \to C$ be a piecewise smooth map from a finite graph $G$ to $C$ such that every vertex has degree 1 or 3, each edge is smoothly embedded, the vertex set is mapped injectively, and at each degree-3 vertex the image lies in the interior of the image of a small neighborhood. Then for every integer $m \geq 2$ there is an $m$-dimensional closed connected regular real algebraic manifold $M \subset \mathbb{R}^{m+1}$, the zero set of a real polynomial, and a real algebraic map $f \colon M \to NInt(C)$ such that $\pi \circ f$ is Morse and its Reeb graph is isomorphic to $G$. The proof builds $M$ by a unit-sphere construction from a region bounded by a polynomial approximation of a regular neighborhood of the embedded graph. The same construction yields explicit versions for the circle in Theorems 2 and 3.

Load-bearing premise

The proof relies on the claim, asserted rather than proved in Section 2, that a small regular neighborhood of the embedded graph can be approximated by the zero set of a real polynomial in the Whitney topology so that the projection to $C$ is Morse and has exactly the prescribed Reeb graph; if this approximation cannot be made, the existence conclusion fails.

Editorial extensions

If this is right

  • For any admissible finite graph, the Reeb graph is realizable by a real algebraic Morse map into any curve with a product collar, in every dimension $m \geq 2$.
  • For the circle, Theorem 2 gives an explicit polynomial degree, $2\sum_{j=1}^{i+1}(a_j-1)+4$, for graphs that are cycles with possibly parallel edges.
  • Theorem 3 shows the same construction can also prescribe the diffeomorphism type of the level sets: along an edge, the fiber is a connected sum of products of spheres determined by chosen integers.
  • The constructed functions admit real algebraic non-singular extensions on compact manifolds, as stated in Theorem 6.
  • These curve-valued results are the first real algebraic reconstruction theorems of this kind, extending the earlier real-line case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the asserted polynomial-approximation step is supplied with a complete proof, Theorem 1 likely holds for any graph admitting a generic embedding into the curve; the obstructions would be exactly the degree and interior conditions.
  • The same unit-sphere construction may generalize to maps into higher-dimensional base spaces by using several squared norms, replacing the Reeb graph by a higher-dimensional Reeb space.
  • The explicit degree bound in Theorem 2 could be tested for sharpness by attempting to realize the same circle-Reeb graphs with polynomials of smaller degree.
  • The circle construction builds a dictionary between configurations of mutually tangent circles in the plane and algebraic hypersurfaces, which could yield further explicit examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the reconstruction of real algebraic maps from closed manifolds onto curves (1-dimensional real algebraic manifolds) with prescribed Reeb graphs. The main result, Theorem 1, asserts that for a graph G admitting a piecewise smooth map c_G:G→C satisfying certain genericity conditions, and for any m≥2, there exist an m-dimensional closed connected regular real algebraic manifold M⊂R^{m+1} and a real algebraic map f:M→NInt(C) such that the composition π_C∘f is Morse and its Reeb graph is isomorphic to G. The proof follows the scheme: embed the graph in a tubular neighborhood, take a small regular neighborhood, approximate its boundary by the zero set of a real polynomial, and apply a 'unit-sphere construction' to produce the manifold and the map. Theorems 2 and 3 give explicit constructions for certain circular graphs, with explicit polynomial degree bounds and, in Theorem 3, control over the diffeomorphism types of preimages. The paper also discusses related work on Reeb graphs of real-valued and circle-valued Morse functions and non-singular extensions.

Significance. If the constructions are correct, the paper extends the author's earlier real-valued reconstruction results to curve-valued real algebraic maps, establishing a new family of explicit real algebraic functions with prescribed Reeb graphs and, in Theorem 3, prescribed preimage diffeomorphism types. The explicit degree bounds in Theorems 2 and 3 are concrete and potentially useful. The unit-sphere construction (Definition 1) is a clean device for converting a planar region with polynomial boundary into a real algebraic manifold. However, the central existence theorem is currently conditional on an unproved approximation step, and the proofs of the explicit theorems contain several assertions that need to be substantiated before the results can be considered established.

major comments (3)
  1. [Section 2, proof of Theorem 1] The decisive step of the proof is asserted rather than proved. Starting with the small regular neighborhood N(G) of the graph, the proof states: 'We can also consider approximating the boundary of this regular neighborhood by the zero set of some real polynomial function in the C^r or C^∞ Whitney topology with r>1 ... by considering the approximation suitably, beforehand.' This is the only bridge between the smooth regular neighborhood and the polynomial f_{c_G,R} whose zero set defines the manifold S_f. For the conclusion, the approximation must simultaneously ensure that {f≥0} is a compact regular neighborhood of the graph, that the composition π_C∘π_{m+1,2}|_{S_f} is Morse, and that its Reeb graph is isomorphic to G with no extra vertices or edges. C^r-closeness of the boundary alone does not imply these properties; a generic polynomial approximation can introduce extra real zeros or extra critical points in the fibers of π_C, which would change the Reeb graph. The cited references do not supply this statement for arbitrary curves: [2] treats Poincaré-Reeb graphs of plane domains with a height function, and [21] is the author's unpublished preprint. Thus Theorem 1 is not established by the proof as written.
  2. [Section 2, proof of Theorem 2] The proof of Theorem 2 is too terse in the verification of the Reeb graph and the Morse property. After constructing the region bounded by the tangent circles, the proof says 'we can have a real algebraic map onto the resulting region like the map presented in the proof of Theorem 1, according to [17] ([21])' and then asserts that the conditions on the values a_j ensure the desired Reeb graph. It is not shown explicitly that the composition π_C∘f has no critical points other than those corresponding to the tangency points, that all critical values are distinct and ordered as in G, and that no additional vertices appear where the inner or outer boundary circles intersect the sector boundaries. These points are load-bearing for the conclusion that the Reeb graph is isomorphic to G.
  3. [Section 2, proof of Theorem 3] The proof of Theorem 3 delegates the final verification to 'exercises on singularity theory' and to references [31] and [22]. Specifically, the identification of the Reeb graph of the composed map and the diffeomorphism types of the preimages is described only by stating that the preimage is 'regarded as a manifold diffeomorphic to the boundary connected sum ...' and that one 'can easily see' the properties. Given the complexity of the iterative unit-sphere construction, a rigorous proof should explicitly locate the singular points of the composition, verify that they correspond exactly to the vertices of G, and show that the preimage of each edge has the asserted diffeomorphism type. This is necessary to support the claims of Theorem 3.
minor comments (5)
  1. [Throughout] The manuscript contains numerous typographical errors and awkward phrasings, e.g., 'algbraic' in reference [29], 'exrecises' in the proof of Theorem 3, 'comact' in reference [6], and the header 'NAOKI KITAZA W A' with an extra space. A careful proofreading is needed.
  2. [Abstract and Introduction] The phrase 'This can be contributed to real algebraic geometry' is unclear; consider rewriting the abstract to state the contribution more directly. Also, the claim that the curve-valued case 'is first considered here' should be qualified to 'in the real algebraic setting,' since differentiable circle-valued cases are discussed in the same paper and in the literature.
  3. [Section 2, Theorem 1 statement] The notation NInt(C) is used both for the interior of N(C) and as the target manifold of f; this is confusing because NInt(C) is an open non-compact manifold while M is closed. The statement should clarify that f maps into an open annulus or strip, and that the image of M is compactly contained in NInt(C).
  4. [Section 2, Definition 1] The definition of the unit-sphere construction refers to 'the paper [17], followed by [21]' but does not state the precise regularity assumptions on F beyond the zero set being a regular algebraic manifold. It would be helpful to state that F has no critical points on its zero set, which is needed for the projection to be special generic, and to give a short justification or precise reference.
  5. [References] Several references are to the author's own preprints, including [21] and [22], which are not published; the text should indicate their status more explicitly and, where possible, state which specific statements are used from them. This is particularly important for the proof of Theorem 1.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found: Theorem 1's construction is not equivalent to its input; the main approximation step is an unproved gap rather than a self-referential derivation.

full rationale

The claimed derivation does not reduce to its inputs by construction. Theorem 1 takes as input a graph G, a curve C with collar N(C), and a generic piecewise smooth map c_G: G -> C; the output is a polynomial-defined manifold S_f and a projection f whose Reeb graph is G. The proof's US-construction is written out explicitly in Definition 1 as S = {(x,y) in R^{k+k'} | F(x) - ||y||^2 = 0}, and the regularity of this set follows from the stated hypotheses on F; it does not assume the target theorem. Self-citations [17] and [21] supply the method, but the relevant equations appear in the paper itself, so they are not load-bearing circular evidence. The only fragile point is the sentence in the proof of Theorem 1: 'We can define our desired map f_{C,N(C)} := pi_{m+1,2}|_{S_{f_{\tilde c_G,R}}} ... in such a way that the resulting Reeb graph ... is isomorphic to G and that the resulting function is a Morse function, by considering the approximation suitably, beforehand.' This asserts that a polynomial approximation of the boundary of a regular neighborhood can be chosen so that the resulting projection is Morse and has exactly Reeb graph G. No theorem establishing this simultaneous genericity is proved or cited in sufficient specificity, and C^r or C^infty closeness alone does not force the exact Reeb graph. This is a completeness gap in the proof of Theorem 1, not a circularity: the approximation is not a fitted parameter, nor is it a renamed version of the conclusion, and Theorems 2 and 3 avoid the gap by constructing their boundary curves explicitly as unions of circles. The self-citations are numerous but not used to assume the target statement, so the circularity score remains low.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The construction imports several standard tools: Whitney approximation, Nash approximation of boundaries by real polynomials, the Reeb graph existence theorem of Saeki, and the unit-sphere construction from the author's prior work. It introduces no fitted data and no invented entities; the only hand-chosen quantities are the small scale parameter and circle radii used to make the disks disjoint and tangent.

free parameters (1)
  • small scale parameter a and radii of tangent circles in Theorems 2 and 3
    Chosen as 'suitable small' or 'suitable' so that the disks are mutually disjoint and tangent; the construction is existential and does not depend on the numeric values, so no fitting is involved.
assumptions (5)
  • standard math Smooth regular neighborhoods exist for subcomplexes of smooth manifolds.
    Used to thicken the embedded graph in the proof of Theorem 1; cited to Hirsch [14].
  • domain assumption Boundaries of compact regions can be approximated by zero sets of real polynomials in the C^r or C^infty Whitney topology.
    Used to obtain the polynomial whose zero set is the boundary of the thickened graph; depends on Nash-Tognoli approximation theory, cited via [23,24].
  • standard math The unit-sphere construction from [17,21] turns a connected region {F >= 0} into a regular real algebraic manifold {(x,y) : F(x) - ||y||^2 = 0} whose canonical projection is special generic.
    This is the core construction, imported from the author's prior work without proof here; it is used in all three theorems.
  • standard math The Reeb space of a smooth function with finite singular values on a closed manifold is a graph.
    Justifies that the quotient space W_c is a Reeb graph; cited to Saeki [32] in Section 1.2.
  • standard math Special generic maps have explicit local normal forms and the topology of their preimages follows from standard singularity theory.
    Used to identify the diffeomorphism types of preimages in Theorems 2 and 3; cited to Golubitsky-Guillemin [13] and Saeki [31].

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Pith. "Pith review of Reconstruction of real algebraic functions into curves with prescribed Reeb graphs." pith.science (2026). https://pith.science/paper/5H44VCDT

@misc{pith2026250709467,
  author       = {Pith},
  title        = {Pith review of: Reconstruction of real algebraic functions into curves with prescribed Reeb graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5H44VCDT}},
  note         = {Machine review of arXiv:2507.09467}
}
read the original abstract

We discuss reconstructing smooth real algebraic maps onto curves whose Reeb graph is as prescribed. This can be contributed to real algebraic geometry, especially in explicit examples in real algebraic geometry in a new way. The Reeb graph of a smooth function is the space of all connected components of preimages of all single points and a natural quotient space of the manifold with the vertex set being all connected components containing some singular points of it. This gives a strong tool in geometry of manifolds and appeared already in 1950 with Morse functions. The Reeb graph of the natural height of the unit sphere of dimension at least 2 is a graph with exactly two vertices and one edge. We reconstruct functions, from general finite graphs, conversely. In the differentiable situations, Sharko pioneered this in 2006, followed by Masumoto-Saeki and Michalak, mainly. Related real algebraic situations have been launched and studied by the author. The curve-valued case is first considered here.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions

    math.AG 2025-08 conditional novelty 4.0 of 10

    Every tree, and certain graphs assembled from single edges and small circles, is the Reeb graph of a Morse-Bott real algebraic function defined by degree-1 and degree-2 polynomials.

Reference graph

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