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Quartic curves and their bitangents. (English) Zbl 1214.14049
This article is a delightful mixture of 19th century algebraic geometry of quartic curves and their modern algorithmic and computational treatment. The years in the list of references are distributed over the period from 1833 until today. For many statements, both old and new references are provided, thus showing the deep roots of the topic as well as its embedding into current research.
The article provides exact algorithms for computing two representations of planar quartic curves in the complex projective plane. The curve is defined by a ternary quartic polynomial $$f(x,y,z)$$ with rational coefficients. The first representation is $f(x,y,z)=\det(xA+yB+zC)$ with symmetric, $$4 \times 4$$-matrices $$A$$, $$B$$, $$C$$. The second is $f(x,y,z)=q_1(x,y,z)^2 + q_2(x,y,z)^2 + q_3(x,y,z)^2$ with quadratic forms $$q_i(x,y,z)$$. For special classes of quartic curves, it can be guaranteed that both representations have real solutions.
The first representation admits 36 inequivalent solutions. One of them is computed from the quartic’s 28 bitangents, the remaining classes are found by implementing a 19th century algorithm by O. Hesse [J. Reine Angew. Math 49, 279–332 (1855; ERAM 049.1317cj)] which is related to Cayley octads, that is, complete intersections of three quadrics in complex projective three space (in general).
The second representation is classified by symmetric $$6 \times 6$$-matrices $$G$$ of rank three such that $$f = v^T \cdot G \cdot v$$ where $$v = (x^2,y^2,z^2,xy,xz,yz)^T$$. The sum of squares representation is obtained from the factorization $$G = H^T \cdot H$$. There are 63 equivalence classes. They can be computed from the quartic’s Steiner complexes (certain sextuples of pairs of bitangents) which in turn are obtained from the determinantal representation.
The article concludes with a study of the set of positive semidefinite matrices $$G$$ of the above type (the Gram spectahedron) and the variety of Cayley octads (which is not directly related to the main topic of this paper but interesting in its own right).
Supplementary material, in particular open source implementations of some of the used algorithms, can be found at http://math.berkeley.edu/~cvinzant/quartics.html.

##### MSC:
 14Q05 Computational aspects of algebraic curves 14H50 Plane and space curves 51N15 Projective analytic geometry 68W30 Symbolic computation and algebraic computation
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##### References:
  Calabi, E., Linear systems of real quadratic forms, Proc. amer. math. soc., 15, 844-846, (1964) · Zbl 0178.35903  Coble, A., ()  Dixon, A.C., Note on the reduction of a ternary quantic to a symmetrical determinant, Cambridge proc., 11, 350-351, (1902) · JFM 33.0140.04  Dolgachev, I., (), Available online atwww.math.lsa.umich.edu/ idolga/topics.pdf  Dolgachev, I.; Ortland, D., Point sets in projective spaces and theta functions, Astérisque, 165, (1988) · Zbl 0685.14029  Dubrovin, B.; Flickinger, R.; Segur, H., Three-phase solutions of the kadomtsev – petviashvili equation, Stud. appl. math., 99, 137-203, (1997) · Zbl 0893.35112  Edge, W.L., Determinantal representations of $$x^4 + y^4 + z^4$$, Math. proc. Cambridge phil. soc., 34, 6-21, (1938) · Zbl 0018.10003  Eisenbud, D.; Popescu, S., The projective geometry of the gale transform, J. algebra, 230, 127-173, (2000) · Zbl 1060.14528  Fulton, W., Algebraic curves, (), reprinted in 1989 · Zbl 0181.23901  Gizatullin, M., On covariants of plane quartic associated to its even theta characteristic, (), 37-74 · Zbl 1121.14024  Greub, W., ()  Harris, J., Galois groups of enumerative problems, Duke math. J., 46, 685-724, (1979) · Zbl 0433.14040  Helton, J.W.; Vinnikov, V., Linear matrix inequality representation of sets, Comm. pure appl. math., 60, 654-674, (2007) · Zbl 1116.15016  Henrion, D., Detecting rigid convexity of bivariate polynomials, Linear algebra appl., 432, 1218-1233, (2010) · Zbl 1183.65023  Hesse, O., Über die doppeltangenten der curven vierter ordnung, J. reine angew. math., 49, 279-332, (1855)  Hilbert, D., Ueber die darstellung definiter formen als summe von formenquadraten, Math. ann., 32, 3, 342-350, (1888) · JFM 20.0198.02  Klein, F., Über den verlauf der abelschen integrale bei den kurven vierten grades, Math. ann., 10, 365-397, (1876) · JFM 08.0302.01  Miller, G.A.; Blichfeldt, H.F.; Dickson, L.E., Theory and applications of finite groups, (1916), J. Wiley & Sons New York, reprinted by Dover Publications Inc., New York, 1961 · Zbl 0098.25103  Nie, J., 2010. Discriminants and nonnegative polynomials, arXiv:1002.2230. Preprint. · Zbl 1245.14061  Nie, J.; Ranestad, K.; Sturmfels, B., The algebraic degree of semidefinite programming, Math. program. ser. A, 122, 379-405, (2010) · Zbl 1184.90119  Plaumann, D., Sturmfels, B., Vinzant, C., 2010. Computing linear matrix representations of Helton-Vinnikov curves. arXiv:1011.6057. Preprint. · Zbl 1328.14093  Plücker, J., Solution d’une question fondamentale concernant la théorie générale des courbes, J. reine angew. math., 12, 105-108, (1834)  Powers, V.; Reznick, B., Notes towards a constructive proof of hilbert’s theorem on ternary quartics, (), 209-227 · Zbl 1026.11044  Powers, V.; Reznick, B.; Scheiderer, C.; Sottile, F., A new approach to hilbert’s theorem on ternary quartics, C. R. math. acad. sci. Paris, 339, 9, 617-620, (2004) · Zbl 1061.11016  Rostalski, P., Sturmfels, B., Dualities in convex algebraic geometry, Rendiconti di Mathematica e delle sue Applicazioni (2011), arXiv:1006.4894 (in press). · Zbl 1234.90012  Salmon, G., A Treatise on the Higher Plane Curves: Intended as a Sequel to “A Treatise on Conic Sections”, 3rd ed., Dublin, 1879; reprinted by Chelsea Publ. Co., New York, 1960. · JFM 05.0340.03  Sanyal, R., Sottile, F., Sturmfels, B., 2009. Orbitopes, arXiv:0911.5436. Preprint. · Zbl 1315.52001  Steiner, J., Eigenschaften der curven vierten grads rücksichtlich ihrer doppeltangenten, J. reine angew. math., 49, 265-272, (1855)  White, H.S., Seven points on a twisted cubic curve, Proc. natl. acad. sci. USA, 8, 464-466, (1915) · JFM 45.0793.02  Vinnikov, V., Self-adjoint determinantal representations of real plane curves, Math. ann., 296, 3, 453-479, (1993) · Zbl 0789.14029  Zeuthen, H.G., Sur LES différentes formes des courbes planes du quatrième ordre, Math. ann., 7, 408-432, (1873) · JFM 06.0367.01
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