Abstract
We consider the problem of providing systems of equations characterizing the forms with complex coefficients that are totally decomposable, i.e., products of linear forms. Our focus is computational. We present the well-known solution given at the end of the nineteenth century by Brill and Gordan and give a complete proof that their system does vanish only on the decomposable forms. We explore an idea due to Federico Gaeta which leads to an alternative system of equations, vanishing on the totally decomposable forms and on the forms admitting a multiple factor. Last, we give some insight on how to compute efficiently these systems of equations and point out possible further improvements.
Emmanuel Briand is supported by the projects MTM2007-64509 (MICINN, Spain) and FQM333 (Junta de Andalucía).
In honor of Professor Federico Gaeta
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References
Emmanuel Briand. Polynômes multisymétriques, 2002. Doctoral thesis. Université de Rennes 1.
Emmanuel Briand. Brill’s equations for the subvariety of factorizable forms. In Laureano González-Vega and Tomás Recio, editors, Actas de los Encuentros de Álgebra Computacional y Aplicaciones: EACA 2004, pages 59–63, 2004. Santander. Available at http://emmanuel.jean.briand.free.fr/publications/
Alexander von Brill. Über symmetrische Functionen von Variabelnpaaren. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-August-Universität zu Göttingen, 20:757–762, 1893.
Jaydeep V. Chipalkatti. Decomposable ternary cubics. Experiment. Math., 11(1):69–80, 2002.
John Dalbec. Multisymmetric functions. Beiträge zur Algebra und Geometrie, 40(1):27–51, 1999.
William Fulton. Young tableaux. London Mathematical Society Student Texts 35. Cambrige University Press, 1997.
Federico Gaeta, 1999. Personal communication.
I.M. Gel’fand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, resultants, and multidimensional determinants. Birkhäuser Boston Inc., Boston, MA, 1994.
Paul Gordan. Das Zerfallen der Curven in gerade Linien. Math. Ann., 45:410–427, 1894.
David Hilbert. Theory of algebraic invariants. Cambridge University Press, Cambridge, 1993. Translated from the German and with a preface by Reinhard C. Laubenbacher, Edited and with an introduction by Bernd Sturmfels.
Michael F. Singer and Felix Ulmer. Linear differential equations and products of linear forms. J. Pure Appl. Algebra, 117/118:549–563, 1997. Algorithms for algebra (Eindhoven, 1996).
Bernd Sturmfels. Algorithms in invariant theory. Texts and Monographs in Symbolic Computation. Springer-Verlag, Wien, New York, 1993.
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Briand, E. (2010). Covariants Vanishing on Totally Decomposable Forms. In: Alonso, M.E., Arrondo, E., Mallavibarrena, R., Sols, I. (eds) Liaison, Schottky Problem and Invariant Theory. Progress in Mathematics, vol 280. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0201-3_14
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DOI: https://doi.org/10.1007/978-3-0346-0201-3_14
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