Geodesic
Topological classification of Morse polynomials
Informatics and Automation, Differential equations and topology. I, Tome 268 (2010), pp. 40-55 
V. I. Arnold. Topological classification of Morse polynomials. Informatics and Automation, Differential equations and topology. I, Tome 268 (2010), pp. 40-55. http://geodesic.mathdoc.fr/item/TRSPY_2010_268_a3/
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     author = {V. I. Arnold},
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     volume = {268},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_268_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The topological classification is discussed for real polynomials of degree 4 in two real independent variables whose critical points and critical values are all different. It is proved that among the 17746 topological types of smooth functions with the same number of critical points, at most 426 types are realizable by polynomials of degree 4.

[1] Arnold V. I., “Smooth functions statistics”, Funct. Anal. and Other Mathematics, 1:2 (2006), 111–118 | DOI | MR | Zbl

[2] Arnold V. I., “Topological classification of Morse functions and generalisations of Hilbert's 16-th problem”, Math. Phys. Anal. and Geom., 10:3 (2007), 227–236 | DOI | MR | Zbl

[3] Arnold V. I., “Topologicheskaya klassifikatsiya trigonometricheskikh mnogochlenov affinnoi gruppy Koksetera $\widetilde A_2$”, Tr. MIAN, 258, 2007, 7–16 | MR | Zbl

[4] Arnold V., “Topological classification of real trigonometric polynomials and cyclic serpents polyhedron”, The Arnold–Gelfand mathematical seminars: Geometry and singularity theory, Birkhäuser, Boston, 1997, 101–106 | DOI | MR | Zbl

[5] Arnold V. I., “Statistika i klassifikatsiya topologii periodicheskikh funktsii i trigonometricheskikh mnogochlenov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 12, no. 1, 2006, 15–24 | MR | Zbl

[6] Arnold V. I., Eksperimentalnoe otkrytie matematicheskikh faktov, Tsikl lektsii, prochitannykh v letnei shkole “Sovremennaya matematika” (Dubna, iyul 2005 g.), MTsNMO, M., 2006