{"id":"c742f133-34a9-48b7-aa1d-8e3078b95e61","arxiv_id":"2507.09467","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For any finite graph satisfying genericity conditions and any dimension at least 2, the paper constructs real algebraic maps to curves whose Reeb graph is isomorphic to the graph.","lead":"The paper gives a construction of smooth real algebraic maps from high-dimensional manifolds to curves so that the Reeb graph, a combinatorial sketch of how level sets change, is any prescribed finite graph. This matters because explicit real algebraic functions with controlled topology are rare, and the result extends the one-dimensional target case to arbitrary curve targets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 rests on an unproved polynomial-approximation step that must preserve the exact Reeb graph; without it the central existence claim is not established.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing gap: the proof of Theorem 1 assumes that a regular neighborhood of the embedded graph can be approximated by the zero set of a real polynomial while preserving Morse-ness and the prescribed Reeb graph. My analysis sharpens the concern by listing the three simultaneous properties the approximation must have and by noting that closeness in the Whitney topology alone is insufficient. The paper's explicit circle-valued constructions (Theorems 2 and 3) are more self-contained and provide partial evidence for the program, but they do not fill the gap in Theorem 1. Therefore the reader's CONDITIONAL verdict is appropriate; my concern confirms it rather than changing it.","tokens_in":14023,"tokens_out":10399,"duration_ms":131636,"concrete_test":"Formulate and prove the missing approximation lemma in the minimal nontrivial case: take the Y graph (one degree-3 vertex and three degree-1 leaves), C = S^1, and m = 2. Choose an explicit smooth regular neighborhood of an embedded Y in the annulus NInt(S^1), then construct a real polynomial f whose zero set is the boundary of this neighborhood and compute the Reeb graph of pi_{S^1} o pi_{3,2}|_{{f(x)=y^2=0}}. If the quotient has extra vertices, or the central vertex has degree different from 3, then the asserted 'suitably beforehand' genericity step fails. If the quotient matches the Y, the check should be repeated with C = R to isolate which part of the generalization from real-valued to curve-valued maps is genuinely unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step in the proof of Theorem 1 (Section 2) is the sentence: \"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^infty Whitney topology with r>1 ... by considering the approximation suitably, beforehand.\" This is the only passage connecting the smooth regular neighborhood N(G) of the embedded graph to the polynomial f_{c_G,R} whose zero set is the boundary of N0(G). For the conclusion, the approximation must simultaneously ensure: (i) {f>=0} is a compact regular neighborhood of the graph with no extra connected components; (ii) after forming S_f = {f(x)-||y||^2=0}, the composition pi_C o pi_{m+1,2}|_{S_f} is Morse; and (iii) its Reeb graph is isomorphic to G, with no extra vertices or edges. C^infty-closeness to the boundary of a smooth neighborhood does not by itself imply (ii) or (iii), and generic polynomial approximation can introduce extra real zeros or extra critical points in the fibers of pi_C. The cited works [2,17,21] do not supply exactly this statement for arbitrary curves: [2] treats Poincare-Reeb graphs of plane domains with a height function, and [21] is an unpublished preprint. Thus Theorem 1 is currently conditional on an unproved approximation lemma; the proof does not establish its existence. Theorems 2 and 3 are less exposed because they construct the boundary directly as an explicit union of circles.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14324,"tokens_out":7795,"duration_ms":91031,"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":[{"comment":"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.","section":"Section 2, proof of Theorem 1"},{"comment":"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.","section":"Section 2, proof of Theorem 2"},{"comment":"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.","section":"Section 2, proof of Theorem 3"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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).","section":"Section 2, Theorem 1 statement"},{"comment":"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.","section":"Section 2, Definition 1"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's central theorem depends on an approximation lemma that is not stated or proved. Given that the author's earlier work and the general approach make the claim plausible, major revision is appropriate. The editor may wish to ask the author to formulate the approximation lemma precisely, state the hypotheses on the polynomial f in terms of the genericity of the projection, and provide a proof that the Reeb graph is preserved under the approximation. The reliance on unpublished preprints [21,22] should also be addressed. The paper is within the scope of the journal, and the explicit constructions in Theorems 2 and 3 are potentially valuable, but the proofs need to be completed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is worth a serious look, but the headline theorem is not proven as written. The genuinely new thing is the curve-valued construction: first real algebraic maps into a curve with prescribed Reeb graphs. Theorems 2 and 3 give explicit circle-valued examples with polynomial degree bounds and prescribed preimage diffeomorphism types, built concretely from unions of tangent circles and the US-construction. I checked the counting and the preimage claims; they hang together. That part is solid and useful.\n\nThe soft spot is Theorem 1. The proof follows the author's real-valued scheme, but the decisive transition from a smooth regular neighborhood of the embedded graph to a real polynomial boundary is one sentence: approximate the boundary 'by considering the approximation suitably, beforehand.' That approximation has to do three jobs at once: stay a regular neighborhood of the graph, keep the projected map Morse, and produce exactly the prescribed Reeb graph with no extra vertices or edges. C^r closeness alone doesn't deliver those, and the cited references do not supply the needed lemma. [2] is about Poincare-Reeb graphs of plane domains; [21] is an unpublished preprint. So Theorem 1 is conditional on an approximation lemma that is neither stated nor proved. This is a real gap, not a nitpick.\n\nThe paper also leans on the author's preprints [19,21,22]. That slows verification but is not itself a defect.\n\nWho is this for: people working in explicit real algebraic geometry or Reeb graphs of algebraic maps. The explicit theorems are the useful part; the general theorem is a plausible conjecture with an incomplete proof.\n\nRecommendation: send to peer review, but with a clear request to prove the approximation lemma or restrict Theorem 1 to explicit cases. I would not cite the general theorem in its current form, but I would cite the construction technique if the paper is accepted.","headline":"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.","tokens_in":14835,"tokens_out":3316,"would_cite":false,"duration_ms":38535,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14P05","14P10","14P20","14P25","57R45","58C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["real algebraic manifolds","real algebraic maps","Reeb graphs","Morse functions","unit-sphere construction","curve-valued functions","special generic maps","real polynomial approximation"],"falsifier":"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.","tokens_in":13809,"feed_emoji":"📐","tokens_out":13117,"duration_ms":131466,"temperature":0.7,"pith_summary":"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.","feed_headline":"Real algebraic maps can realize any prescribed Reeb graph","feed_subtitle":"Every admissible graph, in every dimension at least 2, becomes a polynomial-defined manifold and Morse map.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the polynomial boundary approximation of regular neighborhoods and the graph-realization arguments adapted here.","marker":"[2]"},{"why":"Introduces the unit-sphere construction for real algebraic functions with given Reeb graphs in the real-line case that this paper extends.","marker":"[17]"},{"why":"Gives the real algebraic map onto a connected semi-algebraic region used in the proofs of Theorems 1-3.","marker":"[21]"},{"why":"Surveys the foundational real-algebraic approximation theory used to justify approximating a smooth regular neighborhood boundary by a real algebraic set.","marker":"[23]"},{"why":"Supplies the theory of special generic maps used to describe the canonical projection and the topology of preimages.","marker":"[31]"},{"why":"Establishes that the Reeb space of a smooth function with finitely many singular values is a graph.","marker":"[32]"},{"why":"Provides smooth regular neighborhoods of graphs, the objects later approximated algebraically.","marker":"[14]"},{"why":"Supplies the singularity theory used to verify Morse-ness of the constructed functions.","marker":"[13]"}],"fun_headline_variants":["Every admissible graph is a Reeb graph of a real algebraic map","Admissible graphs are Reeb graphs of real algebraic maps","Real algebraic maps realize any admissible Reeb graph","Prescribed Reeb graphs via real algebraic maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Every admissible graph is a Reeb graph of a real algebraic map","Admissible graphs are Reeb graphs of real algebraic maps","Real algebraic maps realize any admissible Reeb graph","Prescribed Reeb graphs via real algebraic maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000652,"raw_usage":{"total_tokens":2990,"prompt_tokens":944,"completion_tokens":2046,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":560,"completion_tokens_details":{"reasoning_tokens":1981}},"tokens_in":560,"tokens_out":2046,"duration_ms":16451,"temperature":1.0,"reasoning_tokens":1981,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:55:29.219177+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Saeki, Topology of special generic maps of manifolds into Euclidean spaces , Topology Appl","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of special generic maps used to describe the canonical projection and the topology of preimages."},{"cited_title":"Hirsch, Smooth regular neighborhoods , Ann","cited_arxiv_id":null,"evidence_quote":"Provides smooth regular neighborhoods of graphs, the objects later approximated algebraically."}],"review_version":1}