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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].
14P05 Real algebraic sets
14H45 Special algebraic curves and curves of low genus
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