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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:
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