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

SIGMA 8 (2012), 075, 7 pages      arXiv:1210.5318      https://doi.org/10.3842/SIGMA.2012.075
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

Abstract
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.

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