The invariants of the binary decimic

Let O(Vn)SL2 be the algebra of invariants of binary forms of degree n with complex coefficients. The structure of this algebra was already known for n 9. Here the authors study the case n = 10. They show that I = O(V10)SL2 has a minimal set of 106 generators, and give their degrees. Using Hilbert’s characterization of a homogeneous system of parameters as a set that defines the null cone of Vn, that is, the set of binary forms of degree n on which all invariants vanish, the authors find an explicit homogeneous system of 8 parameters for the ring I. This implies that the list of 106 basic invariants is complete. Partial results about I were already known to J. J. Sylvester and F. Franklin [Amer. J. Math. 2 (1879), no. 3, 223–251; MR1505222] and to Tom Hagedorn (unpublished), who found 104 invariants, those of degree up to 19 [cf. P. J. Olver, Classical invariant theory, London Math. Soc. Stud. Texts, 44, Cambridge Univ. Press, Cambridge, 1999; MR1694364 (2001g:13009) (p. 40)]. The remaining two basic invariants of degree 21 seem to appear for the first time in the paper under review.