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Projective $$M$$-cubics and $$M$$-quartics in general position with a maximally intersecting pair of ovals. (English. Russian original) Zbl 0992.14022
St. Petersbg. Math. J. 11, No. 5, 837-852 (2000); translation from Algebra Anal. 11, No. 5, 166-184 (2000).
Let $$C_n$$ denote the set of points of an $$M$$-curve of degree $$n$$ in the real projective plane $$\mathbb{R}\mathbb{P}^2$$. The cubic $$M$$-curve $$C_3$$ consists of two disjoint topological circles $$O_3$$ and $$J_3$$. The oval $$O_3$$ splits $$\mathbb{R}\mathbb{P}^2$$ and the odd branch $$J_3$$ does not split it. The $$M$$-quartic $$C^4$$ consists of four ovals lying outside one another. It is assumed along the paper that one of these ovals, denoted $$O_4$$, meets $$O_3$$ at 12 distinct real points. Consequently, the other three ovals of $$C_4$$ and the odd branch $$J_3$$ have no intersection points, and so they are called free. The main result of the paper under review is a topological classification of the triplets $$(\mathbb{R}\mathbb{P}^2, C_3 \cup C_4,C_3)$$ under the assumptions of transversality of the intersection, the maximality of the number of common real points and the freedom of $$J_3$$ and the three ovals of $$C_4$$ distinct of $$O_4$$. The paper corrects a minor mistake in section VIIIa of a previous article by G. M. Polotovskij [in: Real algebraic geometry, Proc. Conf., Rennes/Fr. 1991, Lect. Notes Math. 1524, 52-74 (1992; Zbl 0789.14028)].

##### MSC:
 14P25 Topology of real algebraic varieties 57M25 Knots and links in the $$3$$-sphere (MSC2010) 14H45 Special algebraic curves and curves of low genus