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On the classification of non-singular curves of degree 8

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Topology and Geometry — Rohlin Seminar

Part of the book series: Lecture Notes in Mathematics ((2179,volume 1346))

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References

  1. Hilbert D. Mathematische Problem.-Arch.Math.Phys. 1901, 2:1, 213–237.

    MathSciNet  MATH  Google Scholar 

  2. Gudkov D.A., Utkin G.A. Topology of curves of 6-th order and surfaces of fourth order (on 16-th Hilbert’s problem).-Gorky. Uch. Zapiski Gork. Univ., v.87, 1969 (Russian). Translated: Amer.Math. Soc.Translations 112, 1978.

    Google Scholar 

  3. Viro O.Y. Curves of degree 7, curves of degree 8 and Ragsdale’s conjecture.-Dokl.Akad.Nauk SSSR, 1980, 254:6, 1306–1310.

    MathSciNet  MATH  Google Scholar 

  4. Viro O.Y. Plane real curves of degrees 7 and 8: new restrictions.-Isv.Akad.Nauk SSSR, ser.Mat, 1983, 47:5, 1135–1150.

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  6. Gudkov D.A. Topology of real projective algebraic varieties.-Usp.Mat.Nauk, 1974, 29:4,178, 3–79.

    MathSciNet  MATH  Google Scholar 

  7. Polotovskiǐ G.M. On the problem of topological classification of positions of ovals of non-singular algebraic curves in the projective plane.-Methods of quolitative theory of diff.equations, Gorky Univ., 1975, v.1, 101–128.

    Google Scholar 

  8. Chislenko Yu.S. M-curves of degree 10.-Zap.Nauchn.Semin. Leningr. Otd.Mat.Inst. Steklova, v.122, 1982, 146–161.

    MathSciNet  MATH  Google Scholar 

  9. Petrovsky I. On the topology of real plane algebraic curves.-Ann. Math., 1938, 39:1, 187–209.

    MathSciNet  Google Scholar 

  10. Harnack A. Über Vieltheiligkeit der ebenen algebraischen Kurven.-Math.Ann., 1876, 10, 189–199.

    Article  MathSciNet  Google Scholar 

  11. Rohlin V.A. Complex topological characteristics of real algebraic curves.-Usp.Mat.Nauk, 1978, 33:5 (203), 77–89.

    MathSciNet  Google Scholar 

  12. Brusotti L. Nucvi metodi construttive di curve piane d’ordine assegnato, dotate del massimo numero di circuiti.I–VI.-Rend.Ist. Lomb., Serie IIa, 1914, 47, 489–504, 797–811; 1915, 48, 182–196; 1916, 49, 495–510, 577–588, 905–919.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Rohlin V.A. Congruences modulo 16 in sixteenth Hilbert’s problem. I.-Funct. Anal.Appl., 1972, 6:4, 58–64.

    MathSciNet  Google Scholar 

  15. Fiedler T. Pencils of lines and topology of real algebraic curves.-Isv.Akad.Nauk SSSR, Ser.Mat., 1982, 46:4, 853–863.

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Gudkov D.A., Krachnov A.D. On periodicity of Euler characteristic of real algebraic (M-1)-manifolds.-Funkt.Anal.Appl., 1973, 7:2, 15–19.

    Article  MathSciNet  Google Scholar 

  18. Kharlamov V.M. New congruences for Euler characteristic of real algebraic varieties.-Funkt.Anal.Appl., 1973, 7:2, 74–78.

    Article  MathSciNet  Google Scholar 

  19. Rohlin V.A. New inequalities in topology of real plane algebraic curves.-Funkt.Anal.Appl., 1980, 14:1, 37–43.

    Article  MathSciNet  Google Scholar 

  20. Fiedler T. New congruences in topology of real plane algebraic curves.-Dokl.Akad.Nauk.SSSR, 1983, 270:1, 56–58.

    MathSciNet  MATH  Google Scholar 

  21. Hilbert D. Über die reellen Züge algebraischer Curven.-Math.Ann. 1891, 38, 115–138.

    Article  MathSciNet  MATH  Google Scholar 

  22. Wiman A. Über die reellen Züge der ebenen algebraischen Kurven.-Math.Ann., 1923, 90, 222–228.

    Article  MathSciNet  MATH  Google Scholar 

  23. Polotovskiǐ G.M. (M-2)-curves of 8-th degree: constructions, open problems.-Gorky Univ., 1984, 194.-Deposited in VINITI 13.02.85, N 1185-85 Dep.

    Google Scholar 

  24. Goryacheva T.V., Polotovskiǐ G.M. Constructions of (M-1)-curves of 8-th degree.-Gorky Univ., 1985.-30-Deposited in VINITI 24.06.85, N 4441-85Dep.

    Google Scholar 

  25. Polotovskiǐ G.M., Tscherbakova A.V. On construction of (M-3)-curves of 8-th degree.-Gorky Univ., 1985-23-Deposited in VINITI 24.06.85, N 4440-85Dep.

    Google Scholar 

  26. Korchagin A.B. New possibilities in Brussotti’s method for construction of M-curves of degree ⩾ 8.-Methods of qualitative theory of differential equations.-Gorky Univ., 1978, 149–159.

    Google Scholar 

  27. Viro O.Y. Gluing algebraic hypersurfaces smoothings of singularities and constructions of curves.-Trudy Leningr. Mezhdunarodn. topolog. conferen., Leningrad, 1983, 149–197.

    Google Scholar 

  28. Shustin E.I. Independence of smoothings of singular points and new M-curves of degree 8.-Usp.Mat.Nauk, 1985, 40:5 (245), 212.

    Google Scholar 

  29. Shustin E.I. New M-and (M-1)-curves of degree 8.-This volume.

    Google Scholar 

  30. Polotovskiǐ G.M. On classification of (M-2)-curves of 8-th degree.-Methods of qualitative theory of diff. equations. Gorky Univ. 1983, 127–138.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  32. Shustin E.I. Counterexamples to Rohlin’s conjecture.-Funkt.Anal. Appl., 1985, 19:2, 94–95.

    Article  MathSciNet  Google Scholar 

  33. Arnold V.I., Varchenko A.N., Gusein-Zade S.M. Singularities of differentiable maps. I., Moscow, Nauka, 1982.

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  35. Polotovskiǐ G.M. Catalog of M-decomposed curves of 6-th degree.-Dokl.Akad.Nauk SSSR, 1977, 236:3, 548–551.

    MathSciNet  Google Scholar 

  36. Polotovskiǐ G.M. Complete classification of M-decomposed curves of 6-th degree in real projective plane.-Gorky Univ., 1978,-103-Deposited in VINITI, 20.04.78, N 1349-78Dep.

    Google Scholar 

  37. Polotovskiǐ G.M. (M-1)-and (M-2)-decomposed curves of 6-th degree.-Methods of qualitative theory of diff. equations.-Gorky Univ. 1978, 130–148 (corrections to this paper see in the book with the same title published in 1980, 217 ("Letter to editor").

    Google Scholar 

  38. Shustin E.I. Hilbert-Rohn method and smoothings of complicated singular points of curves of 8-th degree.-Usp.Mat.Nauk, 1983, 38:6 (234), 158–159.

    MathSciNet  Google Scholar 

  39. Shustin E.I. Real smoothings of simple 5-fold singular point.-Gorky Univ., 1983-Deposited in VINITI 29.12.83, N 6872-83Dep.

    Google Scholar 

  40. Viro O.Y., Kharlamov V.M. Congruences for real algebraic curves with singularities.-Usp.Mat.Nauk, 1980, 35:4 (214), 154–155.

    Google Scholar 

  41. Shustin E.I. Real curves of 8-th degree with a point of four branches quadratically tangent to each other.-Gorky Univ. 1985, deposited in VINITI 15.01.85, N 390-85Dep., 219–244.

    Google Scholar 

  42. Shustin E.I. On application of Hilbert-Rohn method to investigations of curves of 8-th degree.-Gorky Univ, 1983, deposited in VINITI, 23.05.83, N 2745.83Dep., 63–84.

    Google Scholar 

  43. Shustin E.I. New M-curve of degree 8.-Mat.Zametki, 1987, (to appear).

    Google Scholar 

  44. Polotovskiǐ G.M. (M-2)-curves 8-th degree and some conjectures.-Usp.Mat.Nauk, 1981, 36:4 (220), 235–236.

    Google Scholar 

  45. Polotovskiǐ G.M., Shustin E.I. Construction of counterexamples to Rohlin’s conjecture.-Usp.Mat.Nauk, 1984, 39:4 (238), 113.

    Google Scholar 

  46. Polotovskiǐ G.M. Relation between regid isotopy class of a non-singular curve of 5-th degree in ℝP2 and its position with respect to a line.-Funkt.Anal.Appl., 1986, 20:4, 87–88.

    Google Scholar 

  47. Fiedler T. Additional inequalities in topology of real plane algebraic curves.-Isv.Akad.Nauk SSSR, Ser. Matem., 1985, 49:4, 874–883.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute applied mathematics and cybernetics, Gorky State University, Russia

    G. M. Polotovskiǐ

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  1. G. M. Polotovskiǐ
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Oleg Yanovich Viro Anatoly Moiseevich Vershik

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© 1988 Springer-Verlag

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Polotovskiǐ, G.M. (1988). On the classification of non-singular curves of degree 8. In: Viro, O.Y., Vershik, A.M. (eds) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082788

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  • DOI: https://doi.org/10.1007/BFb0082788

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  • Print ISBN: 978-3-540-50237-1

  • Online ISBN: 978-3-540-45958-3

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