Sixty-Four Curves of Degree Six

References

• [Abo et al. 17] H. Abo, A. Seigal, and B. Sturmfels, “Eigenconfigurations of Tensors.” In Algebraic and Geometric Methods in Discrete Mathematics, vol. 685, edited by H. Harrington, M. Omar, and M. Wright, pp. 1–25, Contemporary Mathematics. Providence, RI: American Mathematical Society, 2017.
   Google Scholar
• [Arnol’d 71] V. I. Arnol’d. “The Situation of Ovals of Real Plane Algebraic Curves, the Involutions of Four-Dimensional Smooth Manifolds, and the Arithmetic of Integral Quadratic Forms.” Akademija Nauk SSSR. Funkcional’nyi Analiz i ego Priloženija 5: 3 (1971), 1–9.
   Google Scholar
• [Bochnak et al. 98] J. Bochnak, M. Coste, and M.-F. Roy. “Real Algebraic Geometry.” In Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36. Berlin: Springer-Verlag, 1998.
   Google Scholar
• [Brown 02] C. Brown. Constructing Cylindrical Algebraic Decomposition of the Plane Quickly, Manuscript, 2002.
   Google Scholar
• [Brugallé and López de Medrano 12] E. Brugallé and L. López de Medrano. “Inflection Points of Real and Tropical Plane Curves.” J. Singular. 4 (2012), 74–103.
   Google Scholar
• [Brusotti 21] L. Brusotti. “Sulla “piccola variazione” di una curva piana algebrica reale.” Accademia dei Lincei, Rendiconti 30 (1921), 375–379.
   Google Scholar
• [Bürgisser and Cucker 13] P. Bürgisser and F. Cucker. “Condition. The Geometry of Numerical Algorithms.” In Grundlehren der Mathematischen Wissenschaften, vol. 349. Heidelberg: Springer, 2013.
   Google Scholar
• [Cheng 10] J. Cheng, S. Lazard, L. Peñaranda, M. Pouget, F. Rouillier, and E. Tsigaridas. “On the topology of planar curves.” Math. Comp. Sci. 4 (2010), 113–137.
   Google Scholar
• [Colley and Kennedy 91] S. Colley and G. Kennedy. “A higher-order contact formula for plane curves.” Comm. Algebra 19 (1991), 479–508.
   Web of Science ®Google Scholar
• [Edge 38] W. L. Edge. “Determinantal Representations of x4 + y4 + z4.” Math. Proc. Cambridge Phil. Soc. 34 (1938), 6–21.
   Google Scholar
• [Eigenwillig et al. 07 A. Eigenwillig, M. Kerber, and N. Wolpert. “Fast and Exact Geometric Analysis of Real Algebraic Plane Curves.” Proc. Int. Symp. Symbolic and Algebraic Computation, 2007.
   Google Scholar
• [Gabard 12] A. Gabard. Ahlfors circle maps and total reality: from Riemann to Rohlin, 2012, arXiv:1211.3680.
   Google Scholar
• [Gel’fand et al. 94] I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Boston: Birkhäuser, 1994.
   Google Scholar
• [Fiedler 81] T. Fiedler. “Eine Beschränkung für die Lage von reellen ebenen algebraischen Kurven.” Beiträge zur Algebra und Geometrie 11 (1981), 7–19.
   Google Scholar
• [González-Vega and Necula 02] L. González-Vega and I. Necula. “Efficient Topology Determination of Implicitly Defined Algebraic Plane Curves.” Comp. Aided Geom. Des. 19 (2002), 719–743.
   Web of Science ®Google Scholar
• [Gudkov 69] D. A. Gudkov. “Ovals of Sixth Order Curves,” Gor’kov. Gos.Univ.Uěn.Zap.Vyp. 87 (1969), 14–20.
   Google Scholar
• [Hilbert 91] D. Hilbert. “Über die reellen Züge algebraischer Curven.” Math. Ann. 38 (1891), 115–138.
   Google Scholar
• [Itenberg 94] I. V. Itenberg. “Rigid Isotopy Classification of Curves of Degree 6 with One Nondegenerate Double Point.” Topo. Manif. Varie. Advan. Soviet Math. 18 (1994), 193–208.
   Google Scholar
• [Kalla and Klein 14] C. Kalla and C. Klein. “Computation of the Topological Type of a Real Riemann Surface.” Math. Comp. 83 (2014), 1823–1844.
   Web of Science ®Google Scholar
• [Kharlamov 81] V. M. Kharlamov. “Rigid Classification up to Isotopy of Real Plane Curves of Degree 5.” Funct. Anal. Appl. 15 (1981), 73–74.
   Google Scholar
• [Kharlamov 78] V. M. Kharlamov. “Isotopic Types of Nonsingular Surfaces of Degree 4 in RP3.” Funct. Anal. Appl. 12 (1978), 86–87.
   Google Scholar
• [Kharlamov 84] V. M. Kharlamov. “Classification of Nonsingular Surfaces of Degree 4 in RP3 with Respect to Rigid Isotopies,” Funct. Anal. Appl. 18 (1984), 39–45.
   Google Scholar
• [Kunert and Scheiderer 17] A. Kunert and C. Scheiderer. “Extreme Positive Ternary Sextics.” Trans. Amer. Math. Soc., to appear. doi:https://doi.org/10.1090/tran/7302
   Web of Science ®Google Scholar
• [Lee and Sturmfels 16] H. Lee and B. Sturmfels. “Duality of Multiple Root Loci.” J. Algeb. 446 (2016), 499–526.
   Web of Science ®Google Scholar
• [Lerario and Lundberg 15] A. Lerario and E. Lundberg. “Statistics on Hilbert’s 16th problem.” Int. Math. Res. Notices 12 (2015), 4293–4321.
   Google Scholar
• [Maccioni 17] M. Maccioni. “The Number of REal Eigenvectors of a Real Polynomial.” Boll. Unione Mat. Ital., to appear (2017). arXiv:1606.04737.
   Google Scholar
• [Michalek et al. 17] M. Michalek, H. Moon, B. Sturmfels, and E. Ventura. “Real Rank Geometry of Ternary Forms.” Annali di Matematica Pura ed Applicata 196 (2017), 1025–1054.
   Web of Science ®Google Scholar
• [Nikulin 80] V. V. Nikulin. “Integer Symmetric Bilinear Forms and some of their Geometric Applications.” USSR-Izv. 14 (1980), 103–167.
   Web of Science ®Google Scholar
• [Petrovsky 38] I. Petrovsky. “On the Topology of Real Plane Algebraic Curves.” Ann. of Math. 39 (1938), 189–209.
   Google Scholar
• [Plaumann et al. 11] D. Plaumann, B. Sturmfels, and C. Vinzant. “Quartic Curves and their Bitangents.” J. Symb. Comput. 46 (2011), 712–733.
   Web of Science ®Google Scholar
• [Rohn 13] K. Rohn. “Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fläche 4. Ordnung.” Math. Ann. 73 (1913), 177–229.
   Google Scholar
• [Rokhlin 78] V. A. Rokhlin. “Complex Topological Characteristics of Real Algebraic Curves.” Russ. Math. Surv. 33: 5 (1978), 85–98.
   Google Scholar
• [Seidel and Wolpert 05] R. Seidel and N. Wolpert. “On the Exact Computation of the Topology of Real Algebraic Curves.” Proc. 21st ACM Symp. Comput. Geom., 107–115, 2005.
   Google Scholar
• [Silhol 89] R. Silhol. “Real Algebraic Surfaces.” Lecture Notes in Mathematics, 1392. Berlin: Springer, 1989.
   Web of Science ®Google Scholar
• [Utkin 69] G. A. Utkin. “Construction of Certain Types of Nonsingular fourth Order Surfaces,” Gor’kov. Gos. Univ. Uěn. Zap. Vyp. 87 (1969), 196–211.
   Google Scholar
• [Vervliet et al. 16] N. Vervliet, O. Debals, L. Sorber, M. Van Barel, and L. De Lathauwer. “Tensorlab 3.0.” Available online (http://www.tensorlab.net), March 2016.
   Google Scholar
• [Viro 84] O. Viro. “Gluing of Plane Real Algebraic Curves and Constructions of Curves of Degrees 6 and 7.” In Topology (Leningrad, 1982), Lecture Notes in Math. 1060, pp. 187–200. Berlin: Springer, 1984.
   Web of Science ®Google Scholar
• [Viro 86] O. Viro. “Progress in the Topology of Real Algebraic Varieties over the Last Six Years.” Russ. Math. Surv. 41 (1986) 55–82.
   Web of Science ®Google Scholar
• [Viro 08] O. Viro. “From the Sixteenth Hilbert Problem to Tropical Geometry.” Jap. J. Math. 3 (2008), 185–214.
   Web of Science ®Google Scholar