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.
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.
MSC:
| 14R05 | Classification of affine varieties |
| 14H50 | Plane and space curves |
| 14P25 | Topology of real algebraic varieties |
Keywords:
isotopy types of affine \(M\)-curvesReferences:
| [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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