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

Naoki Kitazawa

REVIEW 3 major objections 4 minor 22 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every finite tree is the Reeb graph of a Morse-Bott real algebraic function whose defining curves are only lines and circles.

desk verdict Plausible new construction with a real gap: the Reeb graph identification is asserted, not proved, and the paper leans heavily on the author's own preprints. read the letter →

arxiv 2508.00729 v2 pith:ZD3IVEFN submitted 2025-08-01 math.AG math.COmath.MG

Naoki Kitazawa This is my paper · ORCID

classification math.AGmath.COmath.MG MSC 05C0505C1014P0514P1014P2557R4558C05

keywords ReebgraphsrealalgebraicfunctionsMorse-Botttreedecompositionssimplecactusmanifoldscanonicalprojection

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 claims that every finite tree can appear as the Reeb graph of a real algebraic Morse-Bott function built from very simple ingredients: straight lines and circles of fixed radius, i.e., zero sets of polynomials of degree 1 or 2. The construction embeds the tree in the plane, surrounds it by such curves, and uses a reconstruction theorem to lift these curves into a real algebraic manifold whose projection back to the plane has the desired Reeb graph. A second theorem extends the result to a class of graphs with simple cactus tree decompositions, where each block is an edge or a circle. If true, this means the topological obstruction to realizing a graph as a Reeb graph in this algebraic setting is essentially the graph's tree-decomposition structure.

What carries the argument

The load-bearing device is Theorem 3, a reconstruction theorem for real algebraic maps whose image is the closure of a non-empty open set $D \subset \mathbb{R}^2$ bounded by real algebraic curves $S_j$. The theorem builds a real algebraic manifold $M$ as the zero set of equations $\prod_{j_I \in \{j \mid ml_1(j)=j'\}} f_{j_I}(x) - \|y_{I,j'}\|^2 = 0$ for each label $j'$, and the canonical projection to $\mathbb{R}^2$ is a real algebraic map onto $D$. Around any point of $D$ the local preimage is a product of disks and spheres, so component counts can be controlled locally; the proof of the main theorem arranges the lines and circles so that these counts change only at levels corresponding to vertices of the tree, making the Reeb graph isomorphic to $G_T$.

What would settle it

Take a concrete small tree, say a path with three edges or a tree with one vertex of degree 3, write out the lines and circles specified by the proof, and compute the number of connected components of the preimage $f^{-1}(t)$ for a dense set of values $t$ between the chosen critical levels. If any value of $t$ shows a component-count change that does not correspond to a vertex of the tree, the Reeb graph would acquire an extra vertex and Theorem 1 would be false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: for any tree $G_T$, there is a real algebraic function obtained as the composition of a real algebraic map with the canonical projection $\pi_{2,1}$, whose defining polynomials $f_j$ have degree 1 or 2, whose each $S_j$ is a circle of fixed radius or a straight line in $\mathbb{R}^2$, which is a Morse-Bott function, and whose Reeb graph is isomorphic to $G_T$. Theorem 4 broadens the statement: certain graphs obtained by replacing selected vertices of the SCT decomposition with circles are also realizable in the same explicit way. The proof works by placing the tree inside a planar region bounded by lines and circles, assigning the curves labels and multiplicities according to Theorem 3, and then reading off the Reeb graph from the component counts of preimages.

Load-bearing premise

The construction assumes that the local preimage bookkeeping around every intersection of the chosen lines and circles works exactly as asserted, so that no unintended critical values appear and the Reeb graph has precisely the vertices and edges of the given tree.

Editorial extensions

If this is right

Every finite tree is realizable as the Reeb graph of a Morse-Bott real algebraic function whose defining polynomials have degree at most 2.
The realizing function is explicit: the construction gives the actual lines and circles, so for any tree the equations are in principle written down.
The class of graphs with simple cactus tree decompositions whose blocks are edges or circles is also realizable, subject to the conditions in Theorem 4.
The result reduces a question about Reeb graphs of algebraic functions to a planar configuration problem about circles and lines.

Reading between the lines

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

A plausible next step, not taken in the paper, is that every graph admitting a simple cactus tree decomposition is realizable, since the paper's obstruction appears to be combinatorial rather than algebraic.
Because the constructed functions are Morse-Bott with only degree-1 and degree-2 polynomials, they are concrete enough to be tested computationally; running the construction on small trees would provide numerical evidence for or against the local bookkeeping assumptions.
If every tree is realizable, then likely no additional topological restriction on Reeb graphs of real algebraic Morse-Bott functions arises from tree structure alone; the same construction may adapt to graphs built from more general polyhedral blocks.
The gap between Morse and Morse-Bott matters: the construction deliberately uses circles to create degenerate singularities, so the result does not immediately say anything about the classical Morse-function realization problem for trees.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper gives a concrete way to realize every tree (and certain cactus-like graphs) as the Reeb graph of a real algebraic Morse-Bott function, with defining curves of degree at most 2. The result is new and the approach is sensible. But the proof of Theorem 1 doesn't actually show the Reeb graph is the desired tree. It states a few bullets about how the added circles intersect the planar region and then says 'by Theorem 3' we're done. The key global topology check—that the region D remains a thickened tree after adding all circles, with slice component counts changing exactly at the vertex levels—is deferred to figures and the author's preprints. I think the stress-test concern is on target: a circle meeting the existing boundary in two points could be a harmless cap, but it could also split a vertical slice or create a hole, either of which adds a branch or cycle to the Reeb graph. The text does not rule that out. Figure 2 makes the intended arrangement clear, but the argument is not written.

I want to give credit where it's due. The construction is explicit and the algebraic data are very concrete. The SCT decomposition framing is a useful way to organize the class of graphs considered. The paper is honest about where the technical details live: Theorem 3, the author's reconstruction theorem from preprints [7] and [9], is the engine, and the paper gives a plausible local sketch of it. The self-citation load is heavy, but that's not inherently a flaw when the cited results are the actual foundation.

The soft spots are in proportion to the paper's ambition. Theorem 4's proof is even more of a sketch. Typos are rampant, but that's not the main issue. The main issue is that the central verification is missing. A specialist can probably fill it in, and I suspect the theorem is true, but the paper as written does not stand alone.

Who is this for? People working on explicit construction of real algebraic maps with prescribed Reeb graphs, and anyone interested in Morse-Bott functions with simple polynomial data. It deserves a serious referee. My recommendation: send to peer review, but require the author to expand the proof of Theorem 1 with a genuine global bookkeeping argument, and either supply the details for Theorem 4 or move it to a separate paper. I would not cite the current arXiv version in my own work until that is done.

Referee Report

3 major / 4 minor

Summary. The paper studies the realization of finite graphs as Reeb graphs of real algebraic functions. Its main result, Theorem 1, asserts that for every finite tree G_T there exists a Morse-Bott real algebraic function, obtained as the composition of a real algebraic map with the canonical projection π_{2,1}, whose Reeb graph is isomorphic to G_T and whose defining polynomials have degree at most 2; the construction uses only straight lines and circles in the plane. Theorem 4 extends the construction to a class of cactus-like graphs built from a tree by replacing certain vertices of the SCT decomposition with circles. The proof is based on a reconstruction theorem, Theorem 3, quoted from the author's preprints, together with a local case analysis in Section 2.2.

Significance. If the construction is completed rigorously, the paper would provide a very explicit geometric realization: every finite tree is the Reeb graph of a Morse-Bott real algebraic function defined using only straight lines and circles, with polynomials of degree at most 2. That would be a concrete and appealing contribution to the study of Reeb graphs of real algebraic functions, going beyond general existence and approximation results. The constructive nature and the precise degree bounds are genuine strengths. However, the significance is conditional: the central identification of the Reeb graph with the prescribed tree is not established in the text, and the paper's main theorem therefore remains a plausible but unproven claim as written.

major comments (3)

[§2.3, proof of Theorem 1] The proof does not establish the global topology of the planar region D, which is the load-bearing step for identifying the Reeb graph with G_T. The bullets describing the remaining circles only impose local intersection conditions; a circle whose disk is deleted from D can split a strip into two components or create a hole while still intersecting the previous union in two points. Since every extra component or 1-cycle in D would appear as an extra branch or cycle in the Reeb graph, the text needs a step-by-step verification that each attached disk is a boundary cap, that D remains a thickened tree, and that the vertical slices D ∩ π_{2,1}^{-1}(t) have component counts prescribed by g_{G_T}. Figure 2 illustrates the intended configuration but does not prove these facts.
[§2.2, Theorem 3 and its use] Theorem 1 reduces to Theorem 3, but Theorem 3 is only sketched here and is quoted from the author's preprints [7,9]. The local Cases 1–3 in the sketch describe normal forms around points of D and its boundary, but they do not verify the global Reeb graph of the resulting function on M. Since Theorem 3 is the central tool and does not by itself state or prove the Reeb-graph identification, the paper needs either a complete proof of Theorem 3 or a direct argument showing that the global Reeb graph of π_{2,1}|_M is the prescribed graph. Without this, the main theorem is not established by the text.
[§2.3, Theorem 4] The proof of Theorem 4 inherits the same gap. For the first case it says only 'we can check' the preimages, and for the second case it asserts that omitting certain circles produces the desired SCT decomposition. No computation shows that the omitted circles convert exactly the prescribed vertices into circles in the Reeb graph and that no other vertices or edges are created. This needs an explicit verification for the same global reasons as Theorem 1.

minor comments (4)

[Abstract and throughout] There are numerous typos and spacing errors ('singl e', 'algbraic', 'W e', 'diﬀerentiable', 'diﬀerent'), and the abstract's sentence about a 'graph with exactly one edge and two edges' is inconsistent with the later 'two vertices' wording; these should be corrected.
[Theorem 3] The notation 'D − D' should be written as the set-theoretic difference of the closure and the open set, e.g., \overline{D} \setminus D; as printed it is confusing. Also, the condition 'D = {x | f_j(x) > 0}' appears to depend on j and should be stated precisely as a single defining inequality system.
[Proof of Theorem 1] The variables l, l1, and I_{g_{G_T},j} are used without a complete list of definitions, and the same quantity appears under different names ('l' and 'l1') in the bullets; a uniform notation would improve readability.
[Figures] Figure 2 is helpful, but the caption does not explain how the colored curves correspond to the subsequent labeling by ml(j); a more detailed caption or a short table of the labels would make the construction easier to follow.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 1's Reeb-graph conclusion is asserted and deferred to the author's preprints; the proof relies on self-citation rather than on an in-text derivation.

self citation load bearing [Section 2.3, proof of Theorem 1, final paragraph]
"By Theorem 3 with a fundamental argument on the singularity theory, explained in [4] for example, and a fundamental argument on real algebraic geometry, explained in [1, 11] (, especially in [11, Discussion 14]), we have a desired real algebraic Morse-Bott function. To know more precise exposition, consult also the preprint [7, 9, 10]."

Theorem 3 is imported from the author's own preprints [7,9], and the paper's only attempt to justify the Reeb graph identification is this final assertion plus a pointer to those preprints. The provided sketch of Theorem 3 (Section 2.2) analyzes only local preimage structure at points of D and never derives the global Reeb graph of π_{2,1}|M; the claim that the Reeb graph equals G_T is therefore not shown inside the paper. It is supported by a self-citation chain rather than by an independent argument, making the central theorem load-bearing on the author's own unreviewed work.

full rationale

There is no equation-level circularity: the paper fits no parameters, defines no concept in terms of its conclusion, and does not rename a known result. The construction in Theorem 1 explicitly places lines and circles around an embedded tree and then applies Theorem 3, which is a general algebraic-geometric reconstruction theorem. However, Theorem 3's stated conclusion concerns only the existence of a real algebraic manifold and a projection; it says nothing about the Reeb graph. The proof of Theorem 1 ends by asserting that a 'desired' Morse-Bott function is obtained and immediately refers the reader to the author's preprints [7,9,10] for a more precise exposition. The included sketch of Theorem 3 covers only local preimage types (Cases 1–3) and does not establish the global component-counting that would prove the Reeb graph is isomorphic to G_T. Thus the central Reeb-graph claim is not derived in the paper itself; it is carried by self-citation to the author's own prior work. The geometric construction is explicit and has independent content, so a score of 4 (some self-citation, central claim still has independent content) is appropriate rather than a higher score. The missing verification of the cap-disk topology is a correctness risk but not a circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The construction uses hand-chosen geometric constants (levels, radii, epsilons) and freely chosen fiber dimensions I_{j'}; these satisfy open conditions (disjointness, transversality) and are not fitted to data. No invented entities in the sense of new particles, forces, or dimensions: the high-dimensional manifold M is an explicit construction, and the 'simple cactus tree decomposition' of Definition 2 is a new combinatorial definition with no independent empirical content. The dominant burden on the ledger is the author's own Theorem 3.

free parameters (3)

Vertex levels and bounding-line positions i_{g_{G_T},j}, Y1, Y2
Chosen generically in the proof of Theorem 1 so that the embedding of the tree sits strictly between the two horizontal lines. These satisfy open conditions, not fitted to data.
Radii r_j and epsilons for the circles S_{4+j} and the remaining circles
Chosen small enough to ensure disjointness and transversality of the circles with the lines and with the tree graph. Hand-chosen, but with a continuum of valid choices.
Fiber dimensions I_1, I_2, I_3, I_4 (with I_4 = 0 in Theorem 4)
Positive integers chosen freely to set the dimension of the constructed manifold M. They control the dimensions of preimage spheres but do not change the Reeb graph.

assumptions (3)

domain assumption Theorem 3 (Reconstruction Theorem): the zero set M defined in Section 2.2 is a non-singular real algebraic manifold, and the projection to R^2 is a real algebraic map with the described local preimage structure.
The main imported tool, from the author's preprints [7] and [9]; re-proved here only as a three-case local sketch. If this theorem fails in a global case, Theorems 1 and 4 fail.
domain assumption For every tree G_T there is a piecewise smooth embedding into R^2 whose projection is a function g_{G_T} whose preimage size grows only by adding components at each level.
Stated in the proof of Theorem 1 as 'a kind of well-known theorems on graphs'; no proof or reference is given for the specific bullet properties.
standard math Standard background: implicit function theorem, Nash-Tognoli approximation, Morse-Bott singularity theory, and Reeb graph finiteness for smooth functions on closed manifolds.
Invoked throughout Sections 1.1, 2.1, and 2.2, citing [1], [3], [4], [11], [15], [17], [19], and [22].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions." pith.science (2026). https://pith.science/paper/ZD3IVEFN

@misc{pith2026250800729,
  author       = {Pith},
  title        = {Pith review of: Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZD3IVEFN}},
  note         = {Machine review of arXiv:2508.00729}
}

abstract

We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of all components of preimages of all single points. Reeb graphs have been fundamental and strong tools in geometry of manifolds since the birth of theory of Morse functions, in the former half of the 20th century. We can easily see that the Reeb graph of the natural height of the unit sphere whose dimension is at least $2$ is a graph with exactly one edge and two edges. We are concerned with realizations of graphs decomposed into trees nicely, each vertex of which corresponds to a graph with exactly one edge and two edges or a graph with exactly two edges homeomorphic to a circle.

Figures

Figures reproduced from arXiv: 2508.00729 by the authors.

**Figure 1.** A piecewise smooth function gGT : GT → R (a piecewise smooth embedding ˜gGT : GT → R 2 ). The preimage of a small open neighborhood of p diffeomorphic to D2 is diffeomorphic to D1×DIj′ +1× Q j ′′∈Nl ′−{j ′}S Ij′′ . The map gives the product map of a Morse function for the natural height of the disk and the identity map on the product of a copy of D1 and finitely many copies of the unit spheres. Case 3. The case p … view at source ↗

**Figure 2.** We present a case with l = 9 , l ′ = 3, and IgGT = 5. The lines Sj are colored in blue (1 ≤ j ≤ 4). We have IgGT ,IgGT −1 = IgGT ,4 = 4. The circles S4+j are colored in red (1 ≤ j ≤ IgGT ,IgGT −1 − 1 = 4 − 1 = 3). The circles SIgGT ,IgGT −1+3+j (1 ≤ j ≤ l − (IgGT ,IgGT −1 + 3) = 9 − (4 + 3) = 9 − 7 = 2) are colored in green. ∗ For a vertex of degree 2, for at least one edge containing the vertex, we can choose a ci… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 12 canonical work pages

[7]

N. Kitazawa, Reconstructing real algebraic maps locally like moment-ma ps with prescribed images and compositions with the canonical projections to t he 1-dimensional real aﬃne space , the title has changed from previous versions, arXiv:2303.1 0723, 2024

work page 2024
[9]

Moment-like maps and real algebraic functions with prescribed preimages

N. Kitazawa, Moment-like maps and real algebraic functions with prescri bed preimages , arXiv:2506.17791

work page Pith review arXiv
[1]

Bochnak, M

J. Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Rela ted Areas (3)], vol. 36, Springer- Verlag, Berlin, 1998. Translated from the 1987 French origi nal; Revised by the authors

work page 1998
[2]

Poincar\'e-Reeb graphs of real algebraic domains

A. Bodin, P. Popescu-Pampu and M. S. Sorea, Poincar´ e-Reeb graphs of real algebraic do- mains, Revista Matem´ atica Complutense, https://link.springer.com/article/10.1007/s13163- 023-00469-y, 2023, arXiv:2207.06871v2

work page Pith review arXiv 2023
[3]

Bott, Nondegenerate critical manifolds , Ann

R. Bott, Nondegenerate critical manifolds , Ann. of Math. 60 (1954), 248–261

1954
[4]

Golubitsky and V

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities , Graduate Texts in Mathematics (14), Springer-Verlag (1974)

work page 1974
[5]

Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol

N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol. Methods in Nonlinear Anal. Vol. 59 No. 2B, 897–912, arXiv:1902.08841

arXiv 1902
[6]

Kitazawa, Real algebraic functions on closed manifolds whose Reeb gra phs are given graphs, Methods of Functional Analysis and Topology Vol

N. Kitazawa, Real algebraic functions on closed manifolds whose Reeb gra phs are given graphs, Methods of Functional Analysis and Topology Vol. 28 No. 4 (2 022), 302–308, arXiv:2302.02339, 2023

arXiv 2023

Show all 22 references

[8]

Kitazawa, Some remarks on real algebraic maps which are topologically special generic maps, submitted to a refereed journal, arXiv:2312.10646

N. Kitazawa, Some remarks on real algebraic maps which are topologically special generic maps, submitted to a refereed journal, arXiv:2312.10646

arXiv
[10]

Kitazawa, Reconstruction of real algebraic functions into curves wit h prescribed Reeb graphs, arXiv:2507.09467

N. Kitazawa, Reconstruction of real algebraic functions into curves wit h prescribed Reeb graphs, arXiv:2507.09467

arXiv
[11]

Koll´ ar,Nash’s work in algebraic geometry , Bulletin (New Series) of the American Matem- atical Society (2) 54, 2017, 307–324

J. Koll´ ar,Nash’s work in algebraic geometry , Bulletin (New Series) of the American Matem- atical Society (2) 54, 2017, 307–324

2017
[12]

Camillo De Lellis, The Masterpieces of John Forbes Nash Jr. , The Abel Prize 2013–2017 (Helge Holden and Ragni Piene, eds.), Springer Internation al Publishing, Cham, 2019, 391– 499, https://www.math.ias.edu/delellis/sites/math.ia s.edu.delellis/ﬁles/Nash Abel 75.pdf, arXiv:1606.02551

2013 arXiv
[13]

Masumoto and O

Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph , Kyushu J. Math. 65 (2011), 75–84

2011
[14]

L. P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold . Topol. Methods in Nonlinear Anal. 52 (2) (2018), 749–762, ar Xiv:1805.06727

2018 arXiv
[15]

Milnor, Lectures on the h-cobordism theorem , Math

J. Milnor, Lectures on the h-cobordism theorem , Math. Notes, Princeton Univ. Press, Prince- ton, N.J. 1965

1965
[16]

Milnor and J

J. Milnor and J. Stasheﬀ, Characteristic classes , Annals of Mathematics Studies, No. 76. Princeton, N. J; Princeton University Press (1974)

1974
[17]

Nash, Real algbraic manifolds , Ann

J. Nash, Real algbraic manifolds , Ann. of Math. (2) 56 (1952), 405–421

1952
[18]

G. Reeb, Sur les points singuliers d´une forme de Pfaﬀ compl´ etement int` egrable ou d´une fonction num´ erique, Comptes Rendus Hebdomadaires des S´ eances de I´Acad´ emiedes Sciences 222 (1946), 847–849

1946
[19]

O. Saeki, Reeb spaces of smooth functions on manifolds , International Mathe- matics Research Notices, maa301, Volume 2022, Issue 11, Jun e 2022, 3740–3768, https://doi.org/10.1093/imrn/maa301, arXiv:2006.0168 9

2022
[20]

Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396

V. Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396

2006
[21]

Steenrod, The topology of ﬁbre bundles , Princeton University Press (1951)

N. Steenrod, The topology of ﬁbre bundles , Princeton University Press (1951)

1951
[22]

Tognoli, Su una congettura di Nash , Ann

A. Tognoli, Su una congettura di Nash , Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 167–185. Osaka Central Advanced Mathematical Institute (OCAMI), 3-3- 138 Sugimoto, Sumiyoshi- ku Osaka 558-8585 TEL: +81-6-6605-3103 Email address : naokikitazawa.formath@gmail.com Webpage: http...

1973

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [7]

N. Kitazawa, Reconstructing real algebraic maps locally like moment-ma ps with prescribed images and compositions with the canonical projections to t he 1-dimensional real aﬃne space , the title has changed from previous versions, arXiv:2303.1 0723, 2024

work page 2024

[2] [9]

Moment-like maps and real algebraic functions with prescribed preimages

N. Kitazawa, Moment-like maps and real algebraic functions with prescri bed preimages , arXiv:2506.17791

work page Pith review arXiv

[3] [1]

Bochnak, M

J. Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Rela ted Areas (3)], vol. 36, Springer- Verlag, Berlin, 1998. Translated from the 1987 French origi nal; Revised by the authors

work page 1998

[4] [2]

Poincar\'e-Reeb graphs of real algebraic domains

A. Bodin, P. Popescu-Pampu and M. S. Sorea, Poincar´ e-Reeb graphs of real algebraic do- mains, Revista Matem´ atica Complutense, https://link.springer.com/article/10.1007/s13163- 023-00469-y, 2023, arXiv:2207.06871v2

work page Pith review arXiv 2023

[5] [3]

Bott, Nondegenerate critical manifolds , Ann

R. Bott, Nondegenerate critical manifolds , Ann. of Math. 60 (1954), 248–261

1954

[6] [4]

Golubitsky and V

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities , Graduate Texts in Mathematics (14), Springer-Verlag (1974)

work page 1974

[7] [5]

Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol

N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with ﬁnitely many singular values , Topol. Methods in Nonlinear Anal. Vol. 59 No. 2B, 897–912, arXiv:1902.08841

arXiv 1902

[8] [6]

Kitazawa, Real algebraic functions on closed manifolds whose Reeb gra phs are given graphs, Methods of Functional Analysis and Topology Vol

N. Kitazawa, Real algebraic functions on closed manifolds whose Reeb gra phs are given graphs, Methods of Functional Analysis and Topology Vol. 28 No. 4 (2 022), 302–308, arXiv:2302.02339, 2023

arXiv 2023