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Moment-like maps and real algebraic functions with prescribed preimages

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that every closed, connected, orientable surface can be realized as the preimage of a single point under a smooth function that is the restriction of a coordinate projection to a non-singular real algebraic manifold, with…

desk verdict The proof of the main theorem fails a transversality condition the paper itself requires: the explicit cylinders are tangent, so the construction as written does not produce the claimed non-singular manifold. read the letter →

arxiv 2506.17791 v1 pith:SUYVS72X submitted 2025-06-21 math.AG math.COmath.GTmath.SG

Naoki Kitazawa This is my paper · ORCID
classification math.AGmath.COmath.GTmath.SG MSC 14P0514P2557R4558C0557R19
keywords realalgebraicmanifoldsmapsMorse-BottfunctionsReebgraphsmoment-likeprescribedpreimagessingularvaluesorientablesurfaces
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 proves that Problem 1 has an affirmative answer whenever the prescribed manifold is a closed, connected, orientable surface. Concretely, for any such surface F and any interval [a,b], the author constructs a real algebraic function with image [a,b], with b as its only singular value, and with the fiber over a diffeomorphic to F. This is a construction result, not merely an existence result: the manifold and the function are given by explicit polynomial equations and coordinate projections. It answers the question in the real algebraic category for all orientable surfaces, generalizing earlier smooth-category constructions and the canonical projection of the sphere.

What carries the argument

The central object is the moment-like map reconstructed in Theorem 1. Starting with a bounded region D in R^n cut out by real polynomial inequalities f_j(x)>0 whose boundary components S_j satisfy a transversality condition, one forms the set M defined by the equations f_i(x)-||y_i||^2=0 in R^n times the product of Euclidean spaces; the theorem asserts M is a non-singular real algebraic manifold and that the canonical projection to R^n is a moment-like map with image D, locally a product of a Morse-Bott function and an identity map. In Theorem 2 this machinery is specialized to n=3, where D is bounded by two planes and a chain of cylinders, and the moment-like map is composed with the projection to one coordinate. The fiber over a is then understood through its Reeb graph, whose first Betti number encodes the genus of the fiber.

What would settle it

For a small case, such as the genus-2 example with l=4, write down the four polynomials defining the two planes and two cylinders, form M by the equations f_i(x)=||y_i||^2, and check that the Jacobian has full rank at every point where several f_i vanish; if any such point fails, M is not a non-singular real algebraic manifold. Alternatively, compute the Reeb graph of the fiber over a for that example and compare its first Betti number with 2.

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Extended reading notes

Core claim

Theorem 2 states that Problem 1 is affirmatively solved in the case F is a closed, connected and orientable surface in m=3. The proof builds a bounded region D in $R^{3}$ whose boundary is formed by two parallel planes and l-3 cylinders, uses Theorem 1 to reconstruct a non-singular real algebraic 3-manifold M from the polynomial equations f_i(x)=||y_i||^2, and then composes the resulting moment-like map with the projection to one coordinate. The fiber over a is shown to be a closed orientable surface whose genus is read off from the Reeb graph of a Morse-Bott function on the fiber: the graph has first Betti number l-2, so by the cited realization criterion the surface has genus l-2. Since l can be chosen arbitrarily, every genus occurs. Theorem 3 gives a companion construction in which a and b are the only two singular values and every level set is diffeomorphic to F.

Load-bearing premise

The construction rests on the claim that the explicitly chosen planes and cylinders can be turned into polynomial functions f_j satisfying the intersection and non-singularity conditions of Theorem 1; the paper asserts this happens 'canonically and suitably' and gives only a sketch of the verification.

Editorial extensions

If this is right

  • For every genus g at least 0 there exists a non-singular real algebraic 3-manifold and a function on it with one singular value whose level set over a is a surface of genus g.
  • The construction is explicit: the manifold is the zero set of polynomial equations of the form f_i(x)=||y_i||^2, and the function is a coordinate projection, so the objects can be written down and checked.
  • The same interval endpoints can be shifted by affine change of coordinates, so any a<b works once one example of each genus exists.
  • Theorem 3 yields a real algebraic function on a 3-manifold with exactly two singular values, both at the ends of the image, and with every regular level set diffeomorphic to a prescribed orientable surface.
  • The result places the earlier smooth-category existence theorems into the real algebraic category for surfaces, showing that the stronger algebraic condition does not obstruct the prescribed-preimage phenomenon in dimension three.

Reading between the lines

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

  • If the transversality verification can be carried out in full detail, the same moment-like reconstruction should adapt to non-orientable surfaces by replacing some cylinders with one-sided hypersurfaces or by modifying the Reeb digraph; the paper does not claim this.
  • The region-and-polynomials template appears adaptable to higher-dimensional targets, as the paper's Problem 3 suggests: replacing the final projection with a projection to R^2 and using cylinders in R^4 could give maps whose fibers over interior points are higher-genus surfaces.
  • A testable consequence is that the genus of the prescribed fiber equals the number of cylinder walls minus two in the chain, so one could predict the level-set topology directly from the combinatorics of D and verify it computationally for small l.
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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 Problem 1: given a closed connected (m-1)-manifold F and real numbers a<b, construct a non-singular real algebraic manifold M in some R^{m+k} and a real algebraic function f:M->R that is the restriction of a canonical projection, has image [a,b], has b as its only singular value, and has f^{-1}(a) diffeomorphic to F. After recalling a theorem from the author's preprint (Theorem 1) that produces 'moment-like maps' from a family of hypersurfaces S_j bounding a region D, subject to transversality and combinatorial conditions, the paper states Theorem 2, which asserts an affirmative answer for all closed connected orientable surfaces when m=3. The proof builds an explicit region D in R^3 bounded by two parallel planes and several cylinders, forms the associated moment-like map, and identifies the preimage of a as an orientable surface of genus l-2 by projecting to the x_1-coordinate and analyzing the Reeb graph. Theorem 3 states a variant with two singular values and prescribed preimage at every point. Section 5 gives additional cases for genus 1, 2, and 3 and discusses open problems.

Significance. If the proof were complete, the paper would provide explicit real algebraic functions with prescribed preimage surfaces, addressing a construction problem that is indeed more difficult than mere existence. The moment-like construction and the use of Reeb-graph classification are promising, and the problem is well motivated. However, the explicit data in Section 4 fail a transversality hypothesis that is essential for Theorem 1, and several key verifications are only asserted. Consequently the main theorem is not established in the present form; the paper needs a corrected construction or a substantially more detailed verification before the result can be accepted.

major comments (3)
  1. [Section 4, proof of Theorem 2 (data S_3 and S_j)] The displayed hypersurfaces S_3: x_2^2+(x_3-a)^2=(b-a)^2 and S_j: (x_1-p_{j-3})^2+x_2^2=(b-a)^2 for j>=4 do not satisfy condition (3a) of Theorem 1. At the points q_j^+=(p_{j-3}, b-a, a) and q_j^-=(p_{j-3}, -(b-a), a), both functions vanish and their gradients are (0, +/-2(b-a), 0), so T_{q_j}S_3 = T_{q_j}S_j and the dimension of the intersection of the two tangent planes is 2, whereas Theorem 1(3a) with l_1'=2 and n=3 requires dimension n-l_1'=1. Thus the statement in the proof that the first and third conditions of Theorem 1 'can be checked easily' is false for the displayed data; the non-singularity of M and the smoothness of the preimage f^{-1}(a) as a surface diffeomorphic to F are consequently not established.
  2. [Section 4, bullet defining S_j] The bullet after the displayed cylinder S_j states that the circle S_j intersect {x_3=a} and the straight line L={(t,0,a)} intersect in a one-point set and have agreeing tangent spaces at that point. This is incompatible with the equation (x_1-p_{j-3})^2+x_2^2=(b-a)^2 in the plane x_3=a: a circle centered on L meets L in two points, and its tangent at such a point is not tangent to L except in a degenerate case. This inconsistency indicates that the intended shape of the region D, and hence the subsequent Reeb-graph computation, is not faithfully represented by the displayed formulas; the geometry needs to be redrawn or the formulas corrected.
  3. [Sections 2 and 4, verification of hypotheses and genus computation] The proof of Theorem 2 depends on assertions that are not demonstrated: in Theorem 1 the non-singularity of M is justified by 'We have checked', and in Section 4 the polynomials f_j are said to be chosen 'canonically and suitably' so that the region D is surrounded by the S_j, with condition (1) and the signs defining D left unspecified. The genus identification also relies on a sketchy Reeb-graph argument and on Figures 1 and 2, with no explicit verification that the projected function on the preimage is Morse-Bott with the claimed numbers of critical points. For a construction paper whose main result is an explicit positive answer, these checks are load-bearing and must be supplied.
minor comments (5)
  1. [Abstract] The abstract contains a typo: 'explcit' should be 'explicit'.
  2. [Sections 4 and 5] The text contains several typos, including 'defired map' (should be 'desired map') and 'stuides' (should be 'studies').
  3. [Section 4, proof of Theorem 2] The notation switches between l_1 and l in the construction; the region uses l_1 hypersurfaces, but the Reeb-graph genus is stated as l-2. The relation between these parameters should be fixed and stated explicitly.
  4. [Theorem 3] Theorem 3 is stated with no proof beyond 'can be checked easily from the proof of Theorem 2'; since it is a separate claimed result, it should either be proved or stated as a corollary with a clear argument.
  5. [Figures 1-4] The proof refers to Figures 1-4 as essential for the region D and the Reeb graph, but the figures are not included in the manuscript text; at minimum the combinatorial description of the Reeb digraph should be given in words, as it is used to compute the genus.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction feeds explicit hypersurface data into Theorem 1 and the surface genus is read off from an external Reeb-graph classification; the Section 4 gap is a transversality concern, not a circular reduction.

full rationale

The derivation chain is not circular. Theorem 2 is obtained by choosing explicit hypersurfaces S_j in R^3, invoking Theorem 1 to build M and the moment-like map, and then reading the genus of the preimage off the Reeb digraph via Gelbukh's external classification [5,6]; the genus is set by the number of cylinders l (genus = l-2), so no quantity is fitted and then renamed as a prediction. Theorem 1 itself is taken from the author's preprint [14], but it is restated in the paper with a proof sketch rather than used as an unexamined black box, and its assumptions do not include the target conclusion. The genuinely weak point is Section 4's assertion that the 1st and 3rd conditions of Theorem 1 can be checked easily; the displayed equations make the transversality of S_3 and S_j questionable, since their tangent planes coincide at the two intersection points. However, a failure of transversality would be a correctness gap in the proof, not a circular step in the logical derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a small number of hand-chosen geometric parameters, the transversality condition, and two external classification theorems. No new physical or unexplained geometric entities are postulated; 'moment-like maps' are a defined class of constructed maps, not an input pulled from nowhere.

free parameters (3)
  • positions s1 and s2 of the affine boundary planes
    Chosen by hand with s2-s1 > 2(l-3)(b-a) to ensure the region D is connected and the Reeb graph has the desired Betti number. Exact values do not affect the genus but are needed for the explicit construction.
  • positions p_j of the boundary cylinders
    Increasing sequence with spacing greater than 2(b-a) and tangency conditions to the line L, chosen to control intersections of boundary hypersurfaces and the resulting Reeb graph.
  • combinatorial data m_{l1,l2} and m_{l2} = l2=2 or 3; m_{l1,l2}(3)=2, m_{l1,l2}(i)=0 or 1, m_{l2}(i)=0
    Manually chosen assignment of disk and sphere factors in the moment-like map reconstruction, determined by the desired preimage topology and the number of boundary cylinders.
assumptions (4)
  • domain assumption Transversality condition (3a) of Theorem 1: for any point in a non-empty intersection of boundary hypersurfaces, the tangent spaces have the expected dimension.
    Required for M to be non-singular and for the local product descriptions to hold. The paper asserts this holds for the explicit choices but does not prove it in detail.
  • standard math Gelbukh's classification of Reeb graphs of Morse-Bott functions on surfaces, references [5] and [6].
    Used to conclude that the constructed Reeb graph forces the preimage surface F to be a closed, connected, orientable surface of genus l-2.
  • standard math Standard real algebraic geometry background from [2,18]: non-singular zero sets of polynomial maps and the implicit function theorem.
    Used implicitly to identify M as a non-singular real algebraic manifold and to justify that the constructed functions are real algebraic.
  • standard math Standard local normal forms for Morse-Bott functions.
    Used to describe the local structure of the moment-like maps and to analyze the topology of the preimages near the singular values.

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Cite this review

Pith. "Pith review of Moment-like maps and real algebraic functions with prescribed preimages." pith.science (2026). https://pith.science/paper/SUYVS72X

@misc{pith2026250617791,
  author       = {Pith},
  title        = {Pith review of: Moment-like maps and real algebraic functions with prescribed preimages},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SUYVS72X}},
  note         = {Machine review of arXiv:2506.17791}
}
read the original abstract

We discuss a problem on singularity theory of differentiable (smooth) or real algebraic maps which is different from knowing existence and has been difficult: constructing explcit real algebraic functions. We discuss construction of real algebraic functions with exactly one singular value, the singular points being of definite type, and prescribed preimages of single points. We discuss generalizations of the canonical projection of the unit sphere around the pole and the Morse-Bott functions around the boundaries of the images with preimages diffeomorphic to the torus. This has been discussed in the differentiable (smooth) category since the pioneering study of Sharko in 2006, followed by Masumoto-Saeki, Michalak and so on: the author has first considered the cases respecting the topologies of the preimages of the points where these studies had not done this essentially. Related real algebraic studies have been started by the author essentially in 2020's and the studies are developing, mainly due to the author.

Figures

Figures reproduced from arXiv: 2506.17791 by the authors.

Figure 1
Figure 1. Our desired region D, restricted to the subspace {(t, t′ , a) | t, t′ ∈ R}, in Theorem 2. Note that the space {(t, t′ , a) | t, t′ ∈ R} is also an affine subspace of R 3 . • Three relations p1 − s1 > b − a, s2 − pl−3 > b − a and pj+1 − pj > 2(b − a) hold. • The cylinder Sj = {(t1, t2, t) | (t1 − pj+3) 2 + t2 2 = (b − a) 2 , t ∈ R} of a circle is defined. If the cylinder is restricted to the circle {(t1, t2, a) | (t1… view at source ↗
Figure 2
Figure 2. Our Reeb digraph for Theorem 2. have local extrema. The first two vertices are of degree 2. The 2(l − 3) vertices are of degree 3 and to each of these vertices exactly one singular point of the function is mapped. Thanks to [5] ([6]) with a kind of fundamental arguments on Morse(-Bott) functions and elementary topological theory of surfaces, F is shown to be a closed, connected and orientable surface of genus l − 2.… view at source ↗
Figure 3
Figure 3. Our desired region D restricted to {(t, t′ , a) | t, t′ ∈ R} (in the l1 = 5 case), in the additional study of Theorem 2. The set {(t, t′ , a) | t, t′ ∈ R} is also an affine subspace of R 3 . We can define the cylinder S4 = {(t1, t2, t) | t ∈ R,(t1, t2) ∈ S4,a} of the uniquely defined straight line S4,a := S4,L,a := {(u s1+s2 2 + (1 − u)s1, u(b − a), a) | u ∈ R} ⊂ {(t, t′ , a) | t, t′ ∈ R} or a circle S4,a := S4,C,a … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Our Reeb digraph in the case l1 = 3: the two blue dots are identified and the two green dots are also identified. change the notation for ”S3” to ”S2”, without changing the subset. We put l2 := 2. We can set ml1,l2 (i) := i for i = 1, 2 and ml2 (i) = 0 for i = 1, 2. We…

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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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