Fundamental systems of binary quantics

Table of the known fundamental systems

For the binary Seventhic exist several diverging pieces of data. The numbers in the first column are Sylvester's (1878), which are computed by tamisage. Already 1882 Hammond showed that these numbers are too small. 1888 von Gall published the numbers in the second row computed by the symbolic method and for a long time it was unknown if they are accurate or just an upper bound. 1985 Dixmier and Lazard computed the exact number of invariants by means of computers. 2002 I also computed the exact number of covariants by means of computers, and an (presumably) almost complete fundamental system of the binary nonic.

With the programme kovariante it was possible to verify the fundamental systems up to degree 6. The fundamental systems of the seventhic could be verified up to order 27 and the one of the octavic up to order 21. Using von Gall's information as upper bound it was possible to prove anew the existence of the invariant of the seventhic in order 30. The notion of Cohen-Macaulayness provides an upper bound on the order of irreducible invariants. These bounds are shown in the table below.

Degree of the binary form	2	3	4	5	6	7	8	9
greatest theoretical order	1	4	3	18	15	48	18	66
greatest real order	1	4	3	18	15	30	10	--

As one can see from the table the highest order of an invariant of the octavic is 18. All irreducible invariants can thus be found by kovariante. A similiar bound for covariants is not known.

On the next pages the complete tables of groundforms are published up to the octavic. The not entirely complete table of groundforms of the binary nonic is added as well.
Columns numbers correspond to the degree in the variables x and y, row numbers to the degree in the coefficients a₀, ..., a_n (this degree is called order). E.g. the quintic posseses an irreducible covariant of degree 1 and order 13.