Moscow Mathematical Journal

Volume 26, Issue 1, January–March 2026 pp. 97–142.

Isotopy Classification of Morse Polynomials of Degree Three on ${\mathbb R}^3$

Authors: V. A. Vassiliev (1)
Author institution:(1) Weizmann Institute of Science, Rehovot, Israel

Summary:

We enumerate all isotopy classes of degree three Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ with nonsingular principal homogeneous parts, proving that there are exactly 37 of them. We also count all 2258 isotopy classes of strictly Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ of degree three with the maximal possible number (eight) of real critical points. A main tool in this classification is a combinatorial computer program that formalizes Morse surgeries, local monodromy and Picard–Lefschetz theory.

2020 Math. Subj. Class. Primary: 14P99; Secondary: 14Q30, 14B07, 32S15.

Keywords: Real algebraic geometry, Morse function, vanishing cycle, topological invariant, versal deformation.

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