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A new affine \(M\)-sextic. (English. Russian original) Zbl 0932.14035
Funct. Anal. Appl. 32, No. 2, 141-143 (1998); translation from Funkts. Anal. Prilozh. 32, No. 2, 91-94 (1998).
From the paper: By an affine \(M\)-curve we mean an affine real algebraic curve \(C\) with the maximum possible number of connected components \((m^2-m+2)/2\), where \(m\) is the degree of \(C\).
A. B. Korchagin and E. I. Shustin [Math. USSR, Izv. 33, No. 3, 501-520 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 6, 1181-1199 (1988; Zbl 0679.14011)] constructed 33 isotopy types of affine \(M\)-curves of degree 6. Other constructions (given in more detail) of these 33 curves are presented by A. B. Korchagin [in: Topology of real algebraic varieties and related topics, Transl., Ser. 2, Am. Math. Soc. 173, 141-155 (1996; Zbl 0858.14029)]. Recently, S. Yu. Orevkov [Topology 38, No. 4, 779-810 (1999; Zbl 0923.14032)] managed to prohibit all isotopy types except for the mentioned 33 types and the types \(A_3(0, 5, 5)^*\), \(A_4(1, 4, 5)^*\), \(B_2(1, 8, 1)\), \(B_2(1, 4, 5)\), and \(C_2(1, 3, 6)^*\), in the notation of the two first cited papers. The present note is devoted to the construction of a curve that realizes \(B_2(1, 8, 1)\). We construct it by a perturbation of a suitable singular rational curve by using the Shustin lemma on the independent smoothing of singularities.

14R05 Classification of affine varieties
14H50 Plane and space curves
14P25 Topology of real algebraic varieties
Full Text: DOI
[1] A. B. Korchagin and E. I. Shustin, Izv. Akad. Nauk SSSR, Ser. Mat.,52, No. 6, 1181–1199 (1988).
[2] A. B. Korchagin, Amer. Math. Soc. Transl. (2),173, 141–155 (1996).
[3] E. I. Shustin, In: Methods of Qualitative Theory and the Theory of Bifurcations [in Russian], Gorky State Univ., Gorky, 1988, pp. 97–105
[4] S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology, to appear. · Zbl 0923.14032
[5] E. I. Shustin, Mat. Zametki,42, No. 2, 180–186 (1987).
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