Quadric, cubic and quartic cones

Rocky Mountain Journal of Mathematics

Volumn 35, Issue 5, 2005, Pages 1627-1656

  Korchagin, Anatoly B ^a   Weinberg, David A ^a  

^a TEXAS TECH UNIVERSITY   (United States)

Author keywords

Topological and isotopy classifications of real algebraic curves

Indexed keywords

EID: 33645136184     PISSN: 00357596     EISSN: None     Source Type: Journal    
DOI: 10.1216/rmjm/1181069654     Document Type: Article

References (48)

1. V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps, Vol. 1, Birkhäuser, Boston, 1985.
2. W.W. Rouse Ball, On Newton's classification of cubic, curves, Proc. London Math. Soc. 50 (1891), 104-143.
3. J.W. Bruce and P.J. Giblin, A stratification of the space of plane quartic curves, Proc. London Math. Soc. 42 (1981), 270-289.
4. D.A. Gudkov, On the topology of real protective algebraic varieties, Uspekhi Mat. Nauk 29 (1974), 3-79;
5. Russian Math. Surveys 29 (1974), 1-79 (in English).
6. _, Plane real projective quartic curves, Lecture Notes in Math., vol. 1346, Springer-Vorlag, Berlin, 1988, pp. 341-347.
7. _, Special forms of fourth order curves, Part 1, Deposited in VINITI, No. 9208-B88, 1 3 (in Russian).
8. _, Special forms of fourth order curves, Part 2, Deposited in VINITI, No. 9207-B88, 1-57 (in Russian).
9. _, Special forms of fourth order curves, Part 3, Deposited in VINITI, No. 6435-B89, 1-67 (in Russian).
10. _, Special forms of fourth order curves, Part 4, Deposited in VINITI, No. 1239-B90, 1-55 (in Russian).
11. _, Special forms of fourth order curves, Part 5, Deposited in VINITI, No. 3847-B90, 1-30 (in Russian).
12. D.A. Gudkov, N.A. Kirsanova and G.F. Nebukina, Points of inflection and double tangents of fourth order curves, Part 1, Deposited in VINITI, No. 4207-82Dep, 1-9 (in Russian).
13. _, Points of inflection and double tangents of fourth order curves, Part 2, Deposited in VINITI, No. 17-83Dep, 1-14 (in Russian).
14. D. A. Gudkov and G.F. Nebukina, Points of inflection and double tangents of fourth order curves, Part 3, Deposited in VINITI, No. 704-84Dep, 1-18 (in Russian).
15. _, Points of inflection and double tangents of fourth order curves, Part 4, Deposited in VINITI, No. 6708-B85, 1-23 (in Russian).
16. _, Points of inflection and double tangents of fourth order curves, Part 5, Deposited in VINITI, No. 6709-B85, 1-17 (in Russian).
17. _, Points of inflection and double tangents of fourth order curves, Part 6, Deposited in VINITI, No. 6710-B85, 1-26 (in Russian).
18. _, Points of inflection and double tangents of fourth order curves, Part 7, Deposited in VINITI, No. 6711-B85, 1-15 (in Russian).
19. _, Double tangents and points of inflection of 4th order curves, Uspekhi Mat. Nauk 39 (1984), 112-113;
20. Russian Math. Surveys 39 (1984) (in English).
21. _, Types and forms of 4th order curves with imaginary singular points, Uspekhi Mat. Nauk 40 (1985), 112-113;
22. Russian Math. Surveys 40 (1984) (in English).
23. _, Real curves of fourth order with imaginary singular points, Deposited in VINITI, No. 1108-B86, 1-22 (in Russian).
24. D.A. Gudkov, G.F. Nebukina and T.I. Tetneva, Special forms of fourth order curves with imaginary singular points, Deposited in VINITI, No. 4374-B88, 1-18 (in Russian).
25. D.A. Gudkov and G.M. Polotovskii, Stratification of the space of the 4th order curves. The joining of stratums, Uspekhi Mat. Nauk 42 (1987), 152;
26. Russian Math. Surveys 42 (1987) (in English).
27. _, Stratification of the space of plane algebraic 4th order curves with respect to algebraic-topological types, Part 1, Deposited in VINITI, No. 5600-B87, 1-55 (in Russian).
28. D.A. Gudkov and G.M. Polotovskii, Stratification of the space of plane algebraic 4th order curves with respect to algebraic-topological types, Part 2, Deposited in VINITI, No. 6331-B90, 1-33 (in Russian).
29. D.A. Gudkov, G.A. Utkin and M.L. Tai, The complete classification of nondecomposing curves of the 4th order, Mat. Sb. 69 (111) (1966), 222-256 (in Russian).
30. A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), 189-198.
31. F. Klein, Eine neue Relation zwischen den Singularitäten einer algebraischen Curve, Math. Ann. 10 (1876), 199-209.
32. _, Gessamelte Mathematische Abhandlungen, Bd. 2, Berlin, 1922.
33. _, Lectures on mathematics, the Evanston Colloquium, Amer. Math. Soc., Providence, RI, 1911, (copyright by Macmillan and Co., 1893).
34. A.B. Korchagin, The 1-st part of Hilbert's 16-th problem: History and main results, Texas Tech Univ., Math. Series 19 (1997), 85-140.
35. A.B. Korchagin and D.A. Weinberg, The isotopy classification of affine quartic curves, Rocky Mountain J. Math. 32 (2002), 255-347.
36. G.F. Nebukina, Forms of curves of the fourth order, Ph.D. Thesis, Lobachevsky Univ. at Nizhny Novgorod 7 (1976). 1994, pp. 402-469.
37. I. Newton, Analysis of the properties of cubic curves and their classification by species, in The mathematical papers of Isaac Newton (D.T. Whiteside, ed.), Cambridge Univ. Press, vol. 2, 1968, pp. 3-89.
38. _, Enumeratio linearum tertij ordinis, in The mathematical paper of Isaac Newton (D.T. Whiteside, ed.), Cambridge Univ. Press, vol. 7, 1976, pp. 565-645.
39. _, The final 'Geometriac libri duo', in The mathematical papers of Isaac Newton (D.T. Whiteside, ed.), Cambridge Univ. Press, vol. 7, 1976, pp. 402-409.
40. F. Nicole, Mémoires de l'Academie Royale des Sciences Année MDCCXXXI (1731), Paris, 1733.
41. V.A. Rokhlin, Complex topological characteristics of real algebraic curves, Uspekhi Mat. Nauk 33 (1978), 77-89;
42. Russian Math. Surveys 33 (1979) (in English).
43. E.I. Shustin, On the varieties of singular algebraic curves, in Methods of qualitative theory of differential equations, Gor'ky State Univ., 1983, 148-163 (in Russian).
44. J. Stirling, Lineæ tetrii ordinis Newtonianæ, 1717.
45. O.Ya. Viro, Generalization of Klein's formula and wave, fronts, Uspekhi Mat. Nauk 36 (1981), 233;
46. Russian Math. Surveys 36 (1981) (in English).
47. D.A. Weinberg, The topological classification of cubic curves, Rocky Mountain J. Math. 18 (1988), 065-679.
48. H.G. Zeuthen, Sur les différents forms des courbes planes du quatrième, ordre, Math. Ann. 7 (1873), 408-432.