A complete system of isotopy invariants is proven for Morse polynomials of degree ≤4 over the reals, with explicit realization of all 71 classes for degree 4 and a count of maximal-critical-point classes up to reflection.
Goryunov, Morsifications of rational functions , in: Topology of real algebraic varieties and related topics
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Isotopy classification of Morse polynomials of degree 4 in ${\mathbb R}^2$
A complete system of isotopy invariants is proven for Morse polynomials of degree ≤4 over the reals, with explicit realization of all 71 classes for degree 4 and a count of maximal-critical-point classes up to reflection.