Geodesic
On the disposition of cubic and pair of conics in a real projective plane. II
Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 61-76.

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The problem of topological classification of real algebraic curves is a classical problem in fundamental mathematics that actually arose at the origins of mathematics. The problem gained particular fame and modern formulation after D. Hilbert included it in his famous list of mathematical problems at number 16 in 1900. This was the problem of classifying curves of the sixth degree, solved in 1969 by D.A. Gudkov [1]. In the same place, Gudkov posed the problem of the topological classification of real algebraic curves of degree 6 decomposing into a product of two non-singular curves under certain natural conditions of maximality and general position of quotient curves. Gudkov’s problem was solved in 1977 by G.M. Polotovsky [2], [3]. At present, after a large series of works by several authors (exact references can be found in [4]), the solution of a similar problem on curves of degree 7 is almost complete. In addition, in [5] a topological classification of curves of degree 6 decomposing into a product of any possible number of irreducible factors in general position, and in [6] a classification of mutual arrangements of $M$-quintics, a couple of lines were found. The present paper is devoted to the case when the irreducible factors of the curve of degree 7 have degrees 3, 2, and 2, and is a continuation of the study begun in [7].
Keywords: non-singular plane real algebraic curves, Hilbert's 16th problem, decomposable curves, topological classification. 
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V. A. Gorskaya. On the disposition of cubic and pair of conics in a real projective plane. II. Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 61-76. http://geodesic.mathdoc.fr/item/CHEB_2022_23_3_a4/

[1] Gudkov D. A., Utkin G. A., “Topologiya krivykh 6-go poryadka i poverkhnostei 4-go poryadka (k 16-i probleme Gilberta)”, Uch. zap. Gorkov. un-ta, 1969, 1–214

[2] Polotovskii G. M., “Katalog M-raspadayuschikhsya krivykh 6-go poryadka”, Dokl. AN SSSR, 236:3 (1977), 548–551 | MR | Zbl

[3] Polotovskii G. M., Polnaya klassifikatsiya $M$-raspadayuschikhsya krivykh 6-go poryadka v veschestvennoi proektivnoi ploskosti, Dep. v VINITI. No 1349-78, M., 1978

[4] Borisov I. M., Polotovskii G. M., “O topologii ploskikh veschestvennykh raspadayuschikhsya krivykh stepeni 8”, Itogi nauki i tekhniki, Sovremennye problemy matematiki. Tematicheskie obzory, 176, 2020, 3–18 | DOI

[5] Kuzmenko T. V., Polotovskiy G. M., “Classification of curves of degree 6 decomposing into a product of $M$-curves in general position”, AMS Translations Ser. 2, 173 (1996), 165–177 | MR | Zbl

[6] Korchagin A. B., Polotovskii G. M., “O raspolozheniyakh ploskoi veschestvennoi kvintiki otnositelno pary pryamykh”, Algebra i analiz, 21:2 (2009), 92–112

[7] Gorskaya V. A., Polotovskii G. M., “O raspolozheniyakh kubiki i pary konik v veschestvennoi proektivnoi ploskosti”, Zhurnal Srednevolzhskogo matematicheskogo obschestva, 22:1 (2020), 24–37

[8] Gudkov D. A., “Topologiya veschestvennykh proektivnykh algebraicheskikh mnogoobrazii”, Usp. mat. nauk, 29:4 (178) (1974), 3–79 | MR | Zbl

[9] Orevkov S. Yu., “Link theory and oval arrangements of real algebraic curve”, Topology, 38 (1999), 779–810 | DOI | MR | Zbl

[10] Orevkov S. Yu., “Clasification flexible $M$-curves of degree 8 up to isotopy”, Geom. Funct. Anal., 12:4 (2002), 723–755 | DOI | MR | Zbl

[11] Lee R., “Algebraic functions and closed braids”, Topology, 22 (1983), 191–202 | DOI | MR | Zbl

[12] Orevkov S. Yu., “Raspolozheniya $M$-kvintiki otnositelno koniki, maksimalno peresekayuschei ee nechetnuyu vetv”, Algebra i analiz, 19:4 (2007), 174–242