Tamisage of binary quantics

Cayley and Sylvester developed the theory of generating functions in order to compute the number of lineary independent covariants of a binary form in each degorder. By cleverly manipulating these generating functions Sylvester was able to "compute" fundamental systems of binary forms for forms of degree 10≤n and n=12. Of course, the groundforms are not computed explicitly; only order, degree and weight can be determined. The tables are complete if and only if Sylvester's fundamental postulate, stating that in a degorder syzygies and groundforms never occur simultaneously, is valid. Unfortunately, this postulate is only valid for binary forms of degree n≤6 and n=8. It has been known for a long time that it is false for n=7 and 9≤n≤14. In my dissertation I prove that it is false for n&ge 15 as well.

The table below displays information for the first 20 fundamental systems. Only the numbers for n≤6 and n=8 are accurate, the other ones are too small, due to the invalidity of the fundamental postulate.

Degree of the binary form	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
# Invarianten	1	1	2	4	5	26	9	89	99	494	109	2.296	1.536	6.339	2.263	29.147	16.854	109.133	29.974
# Kovarianten	2	4	5	23	26	124	69	415	475	1831	948	7.086	5.920	21.098	12.979	82.551	56.540	276.428	128.835
max. degree	2	6	3	9	12	15	18	22	26	30	34	38	42	46	50	54	58	62	66
max. order	2	3	4	18	15	22	12	18	17	20	14	20	17	18	16	20	17	20	17

The full table of groundforms for each binary form can be viewed by using the buttons on the left. Accurate tables are not listed again because they are displayed at one level up -> Grundformen.