Moscow Mathematical Journal

Volume 25, Issue 2, April–June 2025 pp. 249–299.

Isotopy Classification of Morse Polynomials of Degree Four on $\mathbb R^2$

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

Summary:

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 4, 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.

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

Keywords: Real algebraic geometry, Morse function, Milnor fiber, Coxeter–Dynkin graph, vanishing cycle, topological invariant, surgery, Lyashko–Looijenga map.

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