2010/08/31 by Daniel Plaumann, Bernd Sturmfels, Cynthia Vinzant · 4 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Combinatorics #Geometry #Mathematics #Mathematics and Applications #Plane (geometry) #Plane curve #Polynomial and algebraic computation #Projective line #Projective plane #Projective space #Projective test #Pure mathematics #Quartic function #Quartic plane curve #Quartic surface #Twisted cubic #Variety (cybernetics) #cs.SC #math.AG #msc:13P15 #msc:14H45 #msc:14H50 #msc:14Q05
paper · pdf · doi:10.1016/j.jsc.2011.01.007
published as Journal of Symbolic Computation 46 (2011) 712-733 · 26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8, other minor changes
arxiv created 2011/01/10 · openalex publication_date 2011/02/04 · arxiv updated 2012/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.