Sylvester versus Gundelfinger

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.


Introduction
In 1868 Gordan [4] proved that the algebra of invariants of binary forms of given degree is finitely generated. This came as a surprise to Cayley and Sylvester, who had believed that the quintic and septimic had covariant resp. invariant rings that were not finitely generated.
The number of invariants is first infinite in the case of a quantic of the seventh order, or septic; the number of covariants is first infinite in the case of a quantic of the fifth order, or quintic. (Cayley [2]) However, finding a minimal set of generators for these algebras is even today an open problem in all but a few small cases. In the case of V 4 ⊕ V 4 , Gordan found a generating set of size 30, and Sylvester [17] showed that two of these generators are superfluous. He added In 1869 Gundelfinger, a student of Clebsch, wrote a thesis [6] where he constructed generators for the covariants of V 3 ⊕ V 4 'in ordinary symbolic notation', after Clebsch had given him this system as computed by Gordan in his 'obscure' notation (cf. [5, pp. 270-272]). He found 20 generators for the invariants and 64 for the covariants.
Sylvester used the Poincaré series together with his fundamental postulate to show that there could be only 61 independent generators for the covariants of V 3 ⊕ V 4 , and wrote a series of papers [15,18,19,20] showing the superiority of the English methods over the German.
In the first paper Sylvester uses his method (which he calls 'tamisage') to derive the numbers of generators of given degrees in the coefficients of V 4 and those of V 3 , and given order in the variables x, y. The following table is taken from [20]: In the second paper he observes that it follows from the Poincaré series that there are 8 linearly independent covariants of order 2 and multidegree (4,3). Next, he constructs 8 reducible such covariants (products of covariants of lower degree) and argues that these are linearly independent. However, the forms are dependent and only seven are independent. He finishes with the announcement In the third paper he observes that it follows from the Poincaré series that there are 12 linearly independent covariants of order 1 and multidegree (5,4). Next, he constructs 12 reducible such covariants and argues that these are linearly independent. However, the forms are dependent and only eleven are independent. He concludes (false theorem omitted) Here we show that the actual minimal number of generators for the covariants of V 3 ⊕ V 4 is 63. Our results coincide with those of Sylvester and Gundelfinger, with two exceptions: we show that one needs one generating covariant of order 1 and multidegree (5,4), and (only) one generating covariant of order 2 and multidegree (4,3).
For completeness we give the corrected version of Sylvester's table. The two corrected entries are underlined.
for all t ∈ k. In particular, the covariants of V of order 0 are the invariants of V .
Below we shall take k = C, G = SL 2 (k), and V = V n 1 ⊕ · · · ⊕ V np , where V n is the vector space (of dimension n + 1) consisting of 0 and the binary forms of degree n, that is, of the homogeneous polynomials of degree n v(x, y) = a 0 x n + n 1 a 1 x n−1 y + · · · + n n − 1 a n−1 xy n−1 + a n y n , in two variables. This V n is the n-th graded part of k[W ], where W is a 2-dimensional vector space over C with natural action of SL 2 , hence has a natural action of SL 2 .
The main way to construct covariants is via transvectants (Überschiebungen). These are derived from the Clebsch-Gordan decomposition of the SL 2 -module V m ⊗ V n , with m ≥ n: This decomposition defines for each p, 0 ≤ p ≤ n, an h) p , and called the p-th transvectant. It is given explicitly by the following formula: [11,Chapter 5]).
The covariants of V can be identified with the invariants of We identify the covariants of V 3 ⊕ V 4 with the invariants of V 1 ⊕ V 3 ⊕ V 4 and show that a minimal set of generators for the algebra of invariants of this module has size 63.
Doing this type of work requires finding dependencies. Gundelfinger did not try to do this exhaustively, but following Gordan he only noted the obvious ones. Sylvester tried, and made some mistakes, no doubt because he already knew what answer he wanted. For us this is relatively easy -a modern computer has no problems computing the rank of a 40000 by 600000 matrix (which is what is needed in the most straightforward approach).
We had a different problem: up to which degree should we compute covariants or invariants? Gundelfinger 'just' followed Gordan's algorithm, but as far as we know that has not been implemented yet.
The secret knowledge known today but not in the 19th century, is that the ring R of invariants of V 1 ⊕ V 3 ⊕ V 4 (or any such ring) is Cohen-Macaulay (see [9]). It has a homogeneous system of parameters (hsop) j 1 , . . . , j r , algebraically independent, and finitely many further generators i 1 , . . . , i s , such that every invariant can be uniquely written as a linear combination of products i m j m 1 · · · j m h . It follows that the Poincaré series P (t) = d i t i , where d i is the dimension of the degree i part of R, is of the form where the a h and b h are the degrees of the i h and j h .
Corollary 2] for p = 1, and by [1, Theorem 2] in general), so that max h a h − j b j = − i (n i + 1). Therefore, in order to find max h {a h , b h } it suffices to find the b h .

Finding a hsop
Let V(J) stand for the vanishing locus of J. The following result, due to Hilbert, gives a characterisation of homogeneous systems of parameters of k[V n 1 ⊕ · · · ⊕ V np ] SL 2 as sets that define the nullcone of N (V n 1 ⊕ · · · ⊕ V np ): Proposition 1 (Hilbert [8]). Let V = V n 1 ⊕ · · · ⊕ V np , and R = k[V ] SL 2 , and m = n 1 + · · · + n p + p − 3 > 0. A set {j 1 , . . . , j m } of homogeneous elements of R is a system of parameters of R if and only if V(j 1 , . . . , j m ) = N (V ).
But j 2 = j 8 = 0 has no solution. This settles Case 3. By Proposition 1, it follows that these eight invariants form a hsop of the ring of invariants of V 1 ⊕ V 3 ⊕ V 4 .

The degrees of the generators
The Poincaré series of the ring of invariants of V 1 ⊕V 3 ⊕V 4 tells us which is the maximal degree in which we have to look for generators, namely 29. For each i ≤ 29 we do the following: multiply invariants of smaller degrees to see what part of the vector space of invariants of degree i is known. The Poincaré series tells us how big the dimension of this vector space is, and if the known invariants do not yet span this vector space, one constructs in some way further invariants, until they do span. In the following table i denotes the degree of the generators, and d i the number of generators of degree i needed: