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Four $$M$$-curves of degree 8. (English. Russian original) Zbl 1054.14070
Funct. Anal. Appl. 36, No. 1, 76-78 (2002); translation from Funkts. Anal. Prilozh. 36, No. 1, 90-93 (2002).
From the introduction: The author proves the existence of four new topological types of pairs $$(\mathbb{R}\mathbb{P}^2,C_8)$$ (isotopy types of $$C_8$$ in $$\mathbb{R}\mathbb{P}^2)$$, where $$C_d$$ is a smooth real algebraic plane curve of degree $$d$$. The classification of pairs $$(\mathbb{R}\mathbb{P}^2,C_d)$$, constituting a part of Hilbert’s 16th problem, is still far from a good understanding. See e.g. A. B. Korchagin [in: Visiting Scholars’ Lectures 1997, Math. Series 19, Texas Tech. University, 1997, 85–140] which contains the list of the 78 isotopy types previously known to be realizable by $$m$$-curves $$C_8$$ (a curve $$C_d$$ is called an $$M$$-curve if it has the maximal possible number of ovals, which is equal to 22 for $$d=8)$$ and of 13 isotopy types for which such a realizability was unknown.
Theorem 1 changes these numbers to 82 and 9, respectively. The basic statement for constructing $$C_d$$’s with controlled topology is the Viro gluing theorem O. Ya. Viro [in: Topology, general and algebraic topology, and applications, Proc. Int. Conf., Leningrad 1982, Lect. Notes Math. 1060, 187–200 (1984; Zbl 0576.14031); Patchworking real algebraic varieties, Preprint Uppsala University U.U.D.M. Report 1994].
##### MSC:
 14P05 Real algebraic sets 14H45 Special algebraic curves and curves of low genus
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