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arxiv: 2311.11113 · v11 · submitted 2023-11-18 · 🧮 math.AG

Isotopy classification of Morse polynomials of degree 4 in {mathbb R}²

V.A. Vassiliev This is my paper

Pith reviewed 2026-05-24 05:37 UTC · model grok-4.3

classification 🧮 math.AG
keywords Morse polynomialsisotopy classificationinvariantsdegree fourcritical pointsreal algebraic geometry
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The pith

A complete system of invariants distinguishes all 71 isotopy classes of degree-4 Morse polynomials from the plane to the line.

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

The paper introduces a system of invariants designed to distinguish isotopy classes of Morse polynomials mapping the plane to the real line. It proves that this system is complete for all polynomials of degree at most 4, meaning two such polynomials are isotopic precisely when their invariants agree. For degree 4 the system takes 71 distinct values, each realized by an explicit polynomial example. This provides a concrete classification that reduces the problem of comparing two polynomials to checking a finite list of algebraic and topological data.

Core claim

The central claim is that a certain system of invariants of isotopy classes of Morse polynomials R² → R is complete for degrees ≤ 4. For degree four this system admits exactly 71 possible combinations, each of which is realized by a concrete Morse polynomial. In addition the number of classes of strictly Morse degree-four polynomials with the maximal number of real critical points is calculated up to isotopy and reflections in the plane.

What carries the argument

a system of invariants for the isotopy classes of Morse polynomials

If this is right

  • Two Morse polynomials of degree ≤4 are isotopic if and only if they share the same values of these invariants.
  • All 71 combinations of the invariants for degree 4 are attained by actual polynomials.
  • The classification extends to the case of polynomials considered up to reflections in the domain for the strictly Morse case with maximum critical points.
  • The invariants provide a practical method to decide isotopy for low-degree examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be tested on random samples of degree-4 polynomials to verify the count.
  • Similar invariants might apply to Morse functions that are not polynomial.
  • The explicit list allows one to study how the number of classes grows with degree.
  • Connections to the topology of real plane curves defined by the polynomials could be explored.

Load-bearing premise

That the particular system of invariants constructed separates every pair of non-isotopic Morse polynomials when the degree is at most four.

What would settle it

Two explicit degree-4 Morse polynomials that have identical invariants but cannot be deformed into each other by an isotopy preserving the Morse property, or one of the 71 predicted combinations that cannot be realized by any polynomial.

Figures

Figures reproduced from arXiv: 2311.11113 by V.A. Vassiliev.

Figure 1
Figure 1. Figure 1: Caustics for degree 3 polynomials: purse (D + 4 ) and pyra￾mid (D − 4 ) The analogous problem for d = 3 essentially coincides with the enumeration of components of complements of caustics of D4 singularities, see e.g. [8]. Namely, the non-discriminantal principal homogeneous parts of degree three polynomials form two classes D + 4 and D − 4 consisting respectively of polynomials vanishing either on one or … view at source ↗
Figure 2
Figure 2. Figure 2: Standard systems of paths In [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Formal graph of D − 4 singularity virtual Morse functions associated with f by arbitrary chains of virtual surgeries s2, s4, s5, s6 and s7 from Definition 6 (i.e. all surgeries which do not model the collision of critical points). Proposition 1. If the principal homogeneous parts of generic Morse polynomials f, ˜f of the same degree are topologically equivalent (i.e. they vanish on the same non-zero number… view at source ↗
Figure 4
Figure 4. Figure 4: Surgery s6 Theorem 2. In the restriction to generic Morse polynomials with only real critical points, the D-graphs are invariants of isotopy classes of Morse functions, and this invariant is equivalent to the set-valued invariant from §2.2. Proof of this theorem consists of the following two lemmas. Lemma 1. Virtual surgeries that model surgeries of Morse polynomials with only real critical points, without… view at source ↗
Figure 5
Figure 5. Figure 5: D-graph for X + 9 (no local maxima, case A) Proof. In the case of functions with only real critical points, the second element of a virtual Morse function (i.e., the string of intersection indices of vanishing cycles with the set of real points) can be reconstructed from the Morse indices of the corresponding critical points and the intersection indices of these vanishing cycles with each other, see Theore… view at source ↗
Figure 6
Figure 6. Figure 6: D-graph X + 9 (no local maxima, case B) ❜ ❜ ❜ ❜ ❜ ✁ ✛ ✁ ✁ ✁ ✁✕ ❆ ❆ ❆ ❆ ❆❑ ❆ ❆ ❆ ❆ ❆❑ ✁ ✁ ✁ ✁ ✁✕ ✲ [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: D-graph X + 9 (no local maxima, case C) ❞ t ❞ t t ❞ t ❞ ✲ ✛ ✲ ✛ ✲ ✛ ✻ ❄ ✻ ❄ ✻ ❄ ✒ [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: D-graph for X + 9 , one local maximum The D-graph invariant takes three different values on polynomials with passport (M, m+) = (9, 0): they are shown in Figs. 5, 6 and 7. The values of the Card invariant of the polynomials with these D-graphs are 7320, 2460, and 6220, respec￾tively. The set of Morse polynomials with the D-graph shown in [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: D-graph for X1 9 (no local maxima, case A) ❞ t ❞ t ❞ t ❞ tt ✻ ❅ ❅ ❅ ❅ ❅■❅ ✲ [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: D-graph for X1 9 (no local maxima, case B) ❞ t ❞ t ❞ t ❞ tt ✻ ❅ ❅ ❅ ❅ ❅■❅ [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: D-graph X1 9 (no local maxima, case C) ❞ t ❞ t ❞ t ❞ tt ✻❅■❅ ❅❅ ❅❅ ❍❍ ❍❍ ❍ ❍❍ ❍❍ ❍❍ ❍❨ ✲ [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: D-graph X1 9 (no local maxima, case D) For each remaining value of the passport invariant (M, m+) from the list (7), all Morse functions with this value form a single non-empty isotopy class. The values of the Card invariant for them are given in [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: X1 9 , one maximum (left) and two maxima (right) Theorem 4 (X1 9 ). Morse polynomials of degree four with non-discriminantal prin￾cipal homogeneous part vanishing on exactly two real lines can have arbitrary values of the “passport” invariant (M, m+) such that M is equal to 1, 3, 5, 7 or 9, and 0 ≤ m+ < M/2. The D-graph invariant takes exactly four values on polynomials of this type with (M, m+) = (9, 0);… view at source ↗
Figure 14
Figure 14. Figure 14: D-graph of type X2 9 (no local maxima, case A) t t t t t t ❞ ❞ ❞ ✛ ❅ ❅ ❅ ❅ ❅■❅ ✻ ✲ [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: D-graph X2 9 (no local maxima, case B) t t t t t t ❞ ❞ ❞ ✛ ❅ ❅ ❅ ❅ ❅■❅ ✻ ❅ ❅ ❅ ❅ ❅■❅ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁✕ [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: D-graph X2 9 (no local maxima, case C) other by a reflection in any line in R 2 . The sets of polynomials with the D-graphs shown in Figs. 14 and 16 each consist of a single connected component [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: X2 9 , one maximum All statements of the previous paragraph remain true for the polynomials with (M, m+) = (9, 3) if we reverse the orientation of all arrows and replace local minima by maxima and vice versa. Polynomials with a principal part of type X2 9 and (M, m+) = (9, 1) (respecti￾vely, (9, 2)) form a single connected component, their D-graph is shown in [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Simplest Reeb graph for polynomials of class X + 9 but concerns classes of strictly Morse functions (not allowing equal critical values at different critical points). In particular, its classes are generally not preserved by elementary surgeries of type s2. On the other hand, the topological equivalence considered there is weaker than the isotopy equivalence. For example, all three classes of polynomials … view at source ↗
Figure 19
Figure 19. Figure 19: Level sets of a polynomial realizing [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The quadratic part (10) of the Taylor expansion of our function at its cusp point is positive, so this function grows in the locally larger component of the complement of this curve at the cusp point, and so we have the left-hand topological shape of [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 20
Figure 20. Figure 20: Cuspidal ovals r r r r r r [PITH_FULL_IMAGE:figures/full_fig_p030_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Level sets for a realization of [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Level sets of a Morse function with D-graph of [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Perturbations of X + 9 singularities with few critical points this zero level set bounds four different domains of negative values, each of which contains exactly one point of local minimum. Proposition 9. Two polynomials constructed in two previous paragraphs belong to different connected components of the space of Morse polynomials of degree four with strictly positive principal homogeneous parts. Proof… view at source ↗
Figure 24
Figure 24. Figure 24: Level sets of perturbations towards the D-graph of [PITH_FULL_IMAGE:figures/full_fig_p041_24.png] view at source ↗
Figure 12
Figure 12. Figure 12 [PITH_FULL_IMAGE:figures/full_fig_p041_12.png] view at source ↗
Figure 25
Figure 25. Figure 25: Polynomials with two asymptotes, having both minima and maxima Proof. By Theorem 4, the D-graphs of Morse perturbations of X1 9 singularities without local maxima can have only four shapes shown in Figs. 9, 10, 11, and 12. Only the last of these contains a subgraph equivalent to the standard Coxeter￾Dynkin graph of type E7, which necessarily occurs in the D-graph of the polynomial ϕ˜. 4.5. Proof of Propos… view at source ↗
Figure 26
Figure 26. Figure 26: Morse functions of X1 9 type with few critical points Obviously, there are polynomials of each of these three types that are invariant under some reflections. Therefore, each of these types is represented by a single connected component of the space of Morse polynomials. 4.7. Polynomials with two asymptotes and fewer than nine real critical points. By the proved part of Theorem 4, the passports (m−, m×, m… view at source ↗
Figure 27
Figure 27. Figure 27: Level sets for a function realizing [PITH_FULL_IMAGE:figures/full_fig_p046_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Critical level sets for a polynomial of class D − 6 + A3 and its morsification The other critical point of (26) is (x, y) = − 1 3a 2 , 0  . In the local coordinates x˜ ≡ a(x + 1 3a 2 ) + y and y, this function is −1 2 23 4a 5 + 1 6a 3 x˜ 2 + ˜x  − 2 3a x˜ 2 + 1 a 2 y 2  + 1 4a y 4 + ˜x  2 a y 3 + ˜x  − 3 a y 2 + 3 4a x˜ 2  . Its principal quasihomogeneous part is 1 6a 3 x˜ 2 + 1 a 2 xy˜ 2 + 1 4a y … view at source ↗
Figure 29
Figure 29. Figure 29: Realization of [PITH_FULL_IMAGE:figures/full_fig_p049_29.png] view at source ↗
read the original abstract

We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$, prove its completeness for polynomials of degrees $\leq 4$, calculate all 71 possible values of these invariants for the case of degree four, and realize them by concrete Morse polynomials. Also we calculate the number of classes (up to isotopy and reflections in ${\mathbb R}^2$) of strictly Morse polynomials of degree four with the maximal possible number of real critical points.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript introduces a system of invariants for isotopy classes of Morse polynomials R² → R, proves completeness of this system for degrees ≤4 via reduction to finitely many combinatorial types, enumerates all 71 attainable invariant combinations for degree 4, supplies explicit polynomial realizations for each, and computes the number of classes up to isotopy and reflections for strictly Morse degree-4 polynomials attaining the maximum number of real critical points.

Significance. If the result holds, the work delivers a complete, constructive classification of isotopy classes for low-degree Morse polynomials, with the explicit realizations providing direct, falsifiable evidence that each listed invariant tuple is attained. The completeness argument rests on exhaustive enumeration of admissible configurations rather than an unverified general principle, which strengthens the claim for degrees ≤4 and offers a template for similar low-degree classifications in real algebraic geometry.

Simulated Author's Rebuttal

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We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

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No significant circularity detected

full rationale

The paper introduces an explicit system of invariants (critical point indices, critical value orderings, and Reeb graph combinatorial data) for isotopy classes of Morse polynomials. Completeness for degrees ≤4 is established by reduction to a finite list of combinatorial types whose isotopy is controlled directly by these invariants, followed by exhaustive enumeration of the 71 attainable combinations for degree 4 and explicit polynomial realizations. No equations reduce a claimed prediction to a fitted input by construction, no load-bearing self-citation chains are invoked, and the argument is self-contained via direct construction and enumeration rather than external or self-referential principles.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the work relies on standard background facts from Morse theory and isotopy in the plane.

axioms (1)
  • standard math Morse polynomials have only non-degenerate critical points and isotopy preserves this non-degeneracy.
    Invoked implicitly when defining the objects whose isotopy classes are classified.

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    math.AG 2025-10 unverdicted novelty 6.0

    Provides the complete list of connected components of Morse functions in deformations of J10 singularities, finishing the isotopy classification of parabolic real function singularities.

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