Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 075, 7 pages      arXiv:1210.5318
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

Sylvester versus Gundelfinger

Andries E. Brouwer a and Mihaela Popoviciu b
a) Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
b) Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland

Received July 18, 2012, in final form October 12, 2012; Published online October 19, 2012

Let $V_n$ be the ${\rm SL}_2$-module of binary forms of degree $n$ and let $V = V_1 \oplus V_3 \oplus V_4$. We show that the minimum number of generators of the algebra $R = \mathbb{C}[V]^{{\rm SL}_2}$ of polynomial functions on $V$ invariant under the action of ${\rm SL}_2$ equals 63. This settles a 143-year old question.

Key words: invariants; covariants; binary forms.

pdf (379 kb)   tex (88 kb)


  1. Brion M., Invariants de plusieurs formes binaires, Bull. Soc. Math. France 110 (1982), 429-445.
  2. Cayley A., A second memoir upon quantics, Phil. Trans. Royal Soc. London 146 (1856), 101-126.
  3. Derksen H., Kemper G., Computational invariant theory, Encyclopaedia of Mathematical Sciences, Vol. 130, Springer-Verlag, Berlin, 2002.
  4. Gordan P., Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist, J. Reine Angew. Math. 69 (1868), 323-354.
  5. Gordan P., Die simultanen Systeme binärer Formen, Math. Ann. 2 (1870), 227-280.
  6. Gundelfinger S., Zur Theorie des simultanen Systems einer cubischen und einer biquadratischen binären Form, Habilitationsschrift, J.B. Metzler, Stuttgart, 1869.
  7. Hammond J., Note on an exceptional case in which the fundamental postulate of professor Sylvester's theory of tamisage fails, Proc. London Math. Soc. 14 (1882), 85-88.
  8. Hilbert D., Ueber die vollen Invariantensysteme, Math. Ann. 42 (1893), 313-373.
  9. Hochster M., Roberts J.L., Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math. 13 (1974), 115-175.
  10. Morley R.K., On the fundamental postulate of tamisage, Amer. J. Math. 34 (1912), 47-68.
  11. Olver P.J., Classical invariant theory, London Mathematical Society Student Texts, Vol. 44, Cambridge University Press, Cambridge, 1999.
  12. Procesi C., Lie groups. An approach through invariants and representations, Universitext, Springer, New York, 2007.
  13. Springer T.A., Invariant theory, Lecture Notes in Mathematics, Vol. 585, Springer-Verlag, Berlin, 1977.
  14. Springer T.A., On the invariant theory of SU2, Indag. Math. 42 (1980), 339-345.
  15. Sylvester J.J., Détermination du nombre exact des covariants irréductibles du système cubo-biquadratique binaire, C. R. Acad. Sci. Paris 87 (1878), 477-481.
  16. Sylvester J.J., Proof of the hitherto undemonstrated fundamental theorem of invariants, Phil. Mag. 5 (1879), 178-188.
  17. Sylvester J.J., Sur le vrai nombre des covariants élémentaires d'un système de deux formes biquadratiques binaires, C. R. Acad. Sci. Paris 84 (1877), 1285-1289.
  18. Sylvester J.J., Sur le vrai nombre des formes irréductibles du système cubo-biquadratique, C. R. Acad. Sci. Paris 87 (1878), 445-448.
  19. Sylvester J.J., Sur les covariants fundamentaux d'un système cubo-quartique binaire, C. R. Acad. Sci. Paris 87 (1878), 287-289.
  20. Sylvester J.J., Franklin F., Tables of the generating functions and groundforms for the binary quantics of the first ten orders, Amer. J. Math. 2 (1879), 223-251.

Previous article  Next article   Contents of Volume 8 (2012)