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On the classification of decomposing plane algebraic curves

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References

  1. Polotovskiî G.M.Catalog of M-decomposed curves of 6-th degree.-Dokl. Akad. Nauk SSSR, 236:3 (1977), 548–551. (English transl. in Sov. Math. Dokl 18:5(1977), p. 1241–1245.)

    MathSciNet  Google Scholar 

  2. Polotovskiî G.M.Complete classification of M-decomposing curves of 6-th degree in real projective plane.-deposited in VINITI N1349-78Dep,1–103.

    Google Scholar 

  3. Polotovskiî G.M.(M-1)-and (M-2)-decomposing curves of 6-th degree.-Methods of the Qualitative Theory of Diff. Equations, Gos. Univ., Gorky (1978), 130–148.

    Google Scholar 

  4. Arnold V.I., Varchenko A.N., Gusein-Zade S.M. Singularities of differentiable mappings, Vol.I, “Nauka”, Moscow, 1982 (English transl., Birkhäuser, 1985.)

    Google Scholar 

  5. Viro O.Y.Gluing of plane real algebraic curves and constructions of curves degrees 6 and 7.-Lect.N.Math.,Vol.1960 (1984), 187–200.

    Article  MathSciNet  MATH  Google Scholar 

  6. Viro O.Y.Gluing algebraic hypersurfaces, smoothings of singularities and constructions of curves.-Trudy Leningr. Mezhdunar. Topol. Confer., Leningrad, 1983, 149–197.

    Google Scholar 

  7. Viro O.Y.Progress in topology of real algebraic varieties over the last six years.-Usp. Mat. Nauk., 41:3 (1986), 45–67.

    MathSciNet  MATH  Google Scholar 

  8. Shustin E.I.The Hilbert-Rohn method and smoothings of complicated singular points of curves of 8-th degree.-Usp. Mat. Nauk.,38:6 (1983), 157–158.

    MathSciNet  Google Scholar 

  9. Shustin E.I.Real smoothings of simple 5-fold singular point.-deposited in VINITI N6872-83Dep, 1–21.

    Google Scholar 

  10. Shustin E.I.The Hilbert-Rohn method and smoothing of singular points of real algebraic curves.-Dokl. Akad. Nauk SSSR, 281:1 (1985), 33–36.

    MathSciNet  Google Scholar 

  11. Shustin E.I.Independence of smoothings of singular points and new M-curves of degree 8.-Usp. Mat. Nauk., 40:5 (1985), 212.

    Google Scholar 

  12. Goryacheva T.V., Polotovskiî G.M.Constructions of (M-1)-curves of 8-th degree.-deposited in VINITI N4441 85Dep, 1–30.

    Google Scholar 

  13. Polotovskiî G.M.(M-2)-curves of 8-th degree: constructions, open problems.-deposited in VINITI N1185-85Dep, 1–194.

    Google Scholar 

  14. Polotovskiî G.M., Tscherbakova A.V.On the construction of (M-3)-curves of 8-th degree.-deposited in VINITI N4440-85dep, 1–23.

    Google Scholar 

  15. Gudkov D.A.Construction of new series of M-curves.-Dokl. Akad. Nauk SSSR, 200:6 (1971), 1269–1272.

    MathSciNet  MATH  Google Scholar 

  16. Polotovskiî G.M.On the classification of (M-2)-curves of order 8.-Selecta Math. Sovietica 9:4 (1990), 403–409.

    Google Scholar 

  17. Rohlin V.A.Complex topological characteristics of real algebraic curves.-Usp. Mat. Nauk., 33:5 (1978), 77–89 (English transl. in Russian Math. Surveys, 33 (1978).)

    MathSciNet  Google Scholar 

  18. Shustin E.I.Counterexamples to Rohlin conjecture.-Funkt. Anal. Appl., 19:2 (1985), 94–95.

    Article  MathSciNet  Google Scholar 

  19. Polotovskiî G.M., Shustin E. I.Construction of counterexamples to Rohlin conjecture.-Usp. Mat. Nauk 39:4 (1984), 113.

    Google Scholar 

  20. Korchagin A.B.New M-curves of degrees 8 and 9.-Dokl. Akad. Nauk SSSR, 306:5 (1989), 1038–1041 (English transl. in Sov. Math. Dokl., 39:3(1989), p. 569–572.)

    MathSciNet  MATH  Google Scholar 

  21. Rohn K.Die Fläche 4 Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung.-Preisschrift der Fürstl. Jablonowskischen Geselschaft, Leipzig, 1886.

    Google Scholar 

  22. Hilbert D.Uber die Gestalt einer Fläche vierter Ordnung.-Gött. Nachrichten (1909), 308–313.

    Google Scholar 

  23. Viro O.Y.Construction of multicomponent real algebraic surfaces.-Dokl. Akad. Nauk SSSR, 248:2 (1979), 279–282.

    MathSciNet  MATH  Google Scholar 

  24. Kharlamov V.M.On the classification of non-singular surfaces of degree 4 in RP3with respect to rigid isotopies.-Funkt. Anal. Appl., 18:1 (1984), 49–56.

    Article  MathSciNet  Google Scholar 

  25. Hilbert D.Ueber die reellen Züge algebraischer Kurven.-Math. Ann., 38 (1891), 115–138.

    Article  MathSciNet  MATH  Google Scholar 

  26. Brusotti L.Sulla “piccola variazione” di una curva piana algebrica reali.-Rend. Rom. Acc. Linc.(5) 30 (1921), 375–379.

    MATH  Google Scholar 

  27. Gudkov D.A.On the topology of algebraic curves on hyperboloid.-Usp. Mat. Nauk., 34:6 (1979), 26–32.

    MathSciNet  MATH  Google Scholar 

  28. Gudkov D.A., Shustin E.I.The classification of non-singular curves of 8-th degree on ellipsoid.-Methods of the Qualitative Theory of Diff.Equations, Gos. Univ.,Gorky (1980), 104–107.

    Google Scholar 

  29. Fiedler T.Eine Beschränkung fur die Lage von reellen ebenen algebraischen Kurven.-Beiträge zur Algebra und Geometrie,11 (1981),7–19.

    MathSciNet  MATH  Google Scholar 

  30. Marin A.Quelques remarques sur les courber algébriques planes reelles.-Publ. Math. Univ. Paris VII, 9 (1980), 51–68.

    Google Scholar 

  31. Polotovskiî G.M.The connection between rigid isotopic class of nonsingular curve of 5-th degree in RP2and its arrangement with respect to line.-Funkt. Anal. Appl., 20:4 (1986), 87–88.

    Google Scholar 

  32. Kharlamov V.M., Viro O.Y.Extensions of the Gudkov-Rohlin congruence.-Lect. N. Math., 1346 (1988), 357–406.

    Article  MathSciNet  MATH  Google Scholar 

  33. Zvonilov V.I.Strengthened Petrovskiî and Arnold inequalities for curves of odd degree.-Funkt. Anal. Appl., 13:4 (1979), 32–39.

    MathSciNet  Google Scholar 

  34. Korchagin A.B., Shustin E.I.Affine curves of degree 6 and smoothings of a non-degenerate sixth order singular point.-Isv. Akad. Nauk SSSR, Ser. Matem., 52:6 (1988), 1181–1199. (English transl. in Math. USSR Isvestiya, 33:3 (1989), p. 501–520.)

    MATH  Google Scholar 

  35. Shustin E.I.On the isotopic classification of affine M-curves of degree 6.-Methods of the Qualitative Theory of Diff. Equations, Gos.Univ., Gorky (1988), 97–105.

    Google Scholar 

  36. Korchagin A.B.On the reduction of singularities and the classification of nonsingular affine sextics.-deposited in VINITI N1107-B (1986), 1–16.

    Google Scholar 

  37. Chevaillier B.A propos des courbes de degree 6.-C. R. Acad. Sci. Paris, Ser. A 302:1 (1986), 33–38.

    MathSciNet  Google Scholar 

  38. Fiedler T.Pencils of lines and the topology of real algebraic curves.-Isv. Akad. Nauk SSSR, Ser.Matem., 46:4 (1982), 853–863. (English transl. in Math. USSR Isvestiya, 21 (1983).)

    MathSciNet  Google Scholar 

  39. Brusotti L.Nuovi metodi costruttivi di curve piano d'ordine assegnato, dotate del massimo numero di circuiti.-Rend. Ist. Lomb.,Ser. IIa, 47(1914)–49(1916).

    Google Scholar 

  40. Chislenko Y.S. M-curves of degree ten.-Zap. nauchn. semin. nLOMI, v (1982), 146–161. (English transl. in J.Soviet Math., 26 (1984),no.1)

    Google Scholar 

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Michel Coste Louis Mahé Marie-Françoise Roy

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© 1992 Springer-Verlag

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Polotovskiî, G.M. (1992). On the classification of decomposing plane algebraic curves. In: Coste, M., Mahé, L., Roy, MF. (eds) Real Algebraic Geometry. Lecture Notes in Mathematics, vol 1524. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084608

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  • DOI: https://doi.org/10.1007/BFb0084608

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