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

V. A. Vassiliev

TL;DR

This work provides a complete isotopy-theoretic classification of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}$ of degree at most $4$ by introducing and proving the sufficiency of three invariants: the passport $(m_-, m_\times, m_+)$, a set-valued/virtual Morse-function invariant, and the D-graph invariant. The authors prove completeness for $d\le 4$, enumerate all invariant values (71 in total for degree four), and realize them with explicit polynomials, including numerous constructions with positive principal parts and various numbers of real critical points. They further classify strictly Morse polynomials of degree four with nine real critical points and relate the resulting isotopy classes to adjacencies among $X_9$-type singularities, providing detailed realizations and obstructions. The work combines Lyashko--Looijenga deformation theory, combinatorial block-graph invariants, and computer-assisted enumeration to deliver a concrete, operable atlas of Morse polynomial isotopy classes with implications for caustics and equisingularity in real singularity theory.

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.

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

TL;DR

This work provides a complete isotopy-theoretic classification of Morse polynomials of degree at most by introducing and proving the sufficiency of three invariants: the passport , a set-valued/virtual Morse-function invariant, and the D-graph invariant. The authors prove completeness for , enumerate all invariant values (71 in total for degree four), and realize them with explicit polynomials, including numerous constructions with positive principal parts and various numbers of real critical points. They further classify strictly Morse polynomials of degree four with nine real critical points and relate the resulting isotopy classes to adjacencies among -type singularities, providing detailed realizations and obstructions. The work combines Lyashko--Looijenga deformation theory, combinatorial block-graph invariants, and computer-assisted enumeration to deliver a concrete, operable atlas of Morse polynomial isotopy classes with implications for caustics and equisingularity in real singularity theory.

Abstract

We introduce a system of invariants of isotopy classes of Morse polynomials , prove its completeness for polynomials of degrees , 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 ) of strictly Morse polynomials of degree four with the maximal possible number of real critical points.
Paper Structure (40 sections, 41 theorems, 69 equations, 29 figures, 8 tables)

This paper contains 40 sections, 41 theorems, 69 equations, 29 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Stating the problem
  3. Interpretation in the local theory of caustics
  4. Atypical adjacencies of multisingularities and equisingularity problem
  5. About the methods
  6. Invariants of isotopy classes of Morse polynomials
  7. Trivial invariant
  8. Set-valued invariant and virtual Morse functions
  9. D-graph invariant
  10. Main theorems
  11. On the proofs of main theorems
  12. On the classification of strictly Morse polynomials
  13. Normalization of $D$-graphs
  14. Realization of polynomials with positive principal parts and prescribed values of invariants
  15. Realization of the D-graph of Fig. \ref{['miss90a']}
  16. ...and 25 more sections

Key Result

Theorem 1

For any pair of simple singularity classes $\Xi$ and $\tilde{\Xi}$ with $\mu(\Xi) + \mu(\tilde{\Xi}) = 9$ and any class of functions $X_9^\ast$$(\ast = +, 1$ or $2)$, we have $\{\Xi + \tilde{\Xi}\}\rightsquigarrow X_9^*$ if and only if "Yes" is written in Table tabadj at the intersection of the row

Figures (29)

  • Figure 1: Caustics for degree 3 polynomials: purse ($D_4^+$) and pyramid ($D_4^-$)
  • Figure 2: Standard systems of paths
  • Figure 3: Formal graph of $D_4^-$ singularity
  • Figure 4: Surgery $s6$
  • Figure 5: D-graph for $X_9^+$ (no local maxima, case A)
  • ...and 24 more figures

Theorems & Definitions (61)

  • Definition 1
  • Definition 2: cf. Jaw2, § 2
  • Theorem 1
  • Corollary 1
  • Definition 3
  • Definition 4: see AGLV2, § V.3
  • Definition 5
  • Remark 1
  • Definition 6
  • Definition 7
  • ...and 51 more