Symbolic computation of degree-three covariants for a binary form (Q970215)

The authors calculate explicitly a basis of the space of degree-three covariants of a binary form of arbitrary degree. There exist a general combinatorial formula for the dimension of these spaces and an algorithm to compute a basis, but few values are known according to the authors. Their result is new to their knowledge and it corrects the incorrect description in [\textit{D. Hilbert}, Theory of algebraic invariants. Cambridge Mathematical Library. Cambridge: Cambridge University Press, (1933; Zbl 0801.13001)]. Also a basis for the complementary of the subspace of reducible covariants is computed. The result is obtained via the so-called classical symbolic method, which the authors describe in Section 2, and matrix algebra. The covariants of degrees 1 and 2 are readily classified as an illustration, and in Sections 3 and 4 the tools for the analysis of degree 3 are introduced. The necessary calculations are carried out in sections 5 and 6.