Geodesic
On mutual arrangements of two $M$-curves of degree~$4$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 3, Tome 222 (2023), pp. 69-82.

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We consider the problem of topological classification of mutual arrangements in the real projective plane of two $M$-curves of degree $4$. We study arrangements under the maximality condition (the oval of one of these curves has $16$ pairwise distinct common points with the oval of the other curve) and some combinatorial condition to select a special type of such arrangements. We list pairwise different topological models of arrangements of this type that satisfy the topological consequences of Bezout's theorem. There are more than 2000 such models. Examples of curves of degree $8$ realizing some of these models are given; we prove that 1728 models cannot be realized by curves of degree $8$. Proofs of the nonrealizability are performed out by Orevkov's method based on the theory of braids and links.
Keywords: plane real algebraic curve, quasi-positive braid, Orevkov's method, Murasugi–Tristram inequality, Fox–Milnor condition.
Mots-clés : decomposable curve 
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N. D. Puchkova. On mutual arrangements of two $M$-curves of degree~$4$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 3, Tome 222 (2023), pp. 69-82. http://geodesic.mathdoc.fr/item/INTO_2023_222_a6/

[1] Borisov I. M., Polotovskii G. M., “O topologii ploskikh veschestvennykh raspadayuschikhsya krivykh stepeni $8$”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obz., 176 (2020), 3–18

[2] Gorskaya V. A., Polotovskii G. M., “O raspolozheniyakh kubiki i pary konik v veschestvennoi proektivnoi ploskosti”, Zh. Srednevolzh. mat. o-va., 22:1 (2020), 24–37

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

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

[5] Hilbert D., “Über die reellen Züge algebraischer Curven”, Math. Ann., 38 (1891), 115–138

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

[7] Rudolf L., “Algebraic functions and closed braids”, Topology., 22 (1983), 191–202