Poincaré series for 1 form, for 2 forms, for more forms.

Invariants for a single form, multiple forms.

Numbers of basic invariants and covariants for V+W, given as #inv/#cov.

V \ W V1 V2 V3 V4
V1 1/3
V2 2/5 3/6
V3 4/13 5/15 7/26
V4 5/20 6/18 20/63 8/28
V5 23/94 29/92 50/.
V6 26/135 27/99 60/194

## V3+V1

4 invariants, 13 covariants.

Numbers of covariants with given degree and order:

d\o 0 1 2 3 # cum
1 - 1 - 1 2 2
2 - - 2 - 2 4
3 - 2 - 1 3 7
4 3 - 1 - 4 11
5 - 1 - - 1 12
6 1 - - - 1 13

## V3+V2

5 invariants, 15 covariants.

Numbers of covariants with given degree and order:

d\o 0 1 2 3 # cum
1 - - 1 1 2 2
2 1 1 1 1 4 6
3 1 1 1 1 4 10
4 1 1 - - 2 12
5 1 1 - - 2 14
6 - - - - - 14
7 1 - - - 1 15

## V4+V2

6 invariants, 18 covariants.

Numbers of covariants with given degree and order:

d\o 0 1 2 3 4 5 6 # cum
1 - - 1 - 1 - - 2 2
2 2 - 1 - 2 - - 5 7
3 2 - 2 - 1 - 1 6 13
4 1 - 1 - 1 - - 3 16
5 - - 1 - - - - 1 17
6 1 - - - - - - 1 18

## V4+V3

20 invariants, 63 covariants. See the separate page.

## 2V4

8 invariants, 28 covariants.

Numbers of covariants with given degree and order:

d\o 0 2 4 6 8 # cum
1 - - 2 - - 2 2
2 3 1 3 1 - 8 10
3 4 2 2 4 - 12 22
4 1 3 - - - 4 26
5 - 2 - - - 2 28

The same table, but with multidegrees given:

d\o 0 2 4 6 # cum
1     1.0/0.1   2 2
2 2.0/1.1/0.2 1.1 2.0/1.1/0.2 1.1 8 10
3 3.0/2.1/1.2/0.3 2.1/1.2 2.1/1.2 3.0/2.1/1.2/0.3 12 22
4 2.2 3.1/2.2/1.3     4 26
5   3.2/2.3     2 28

Gordan (1870, p. 275) finds 8 invariants, 30 covariants, and Salmon (1876, pp. 205-206) just quotes this: "Gordan has enumerated the total number of independent forms for the system of two quartics as thirty". Bertini (1876) finds the same result. Sylvester (1877,1879) finds 8 invariants, 28 covariants, and shows that two of Gordan's covariants are not irreducible.

...

Later, d'Ovidio (1880) showed in a different way that these two invariants of Gordan are reducible.

Young (1899) does the general case of pV4.

### References

P. Gordan, Die simultanen Systeme binärer Formen, Math. Ann. 2 (1870) 227-280.

Salmon, Lessons introductory to the modern higher algebra, 3rd ed., Cambridge, 1876.

E. Bertini, Sistema simultaneo di due forme biquadratiche binarie, Giorn. di mat. 14 (1876) 1-14; translated as Système simultané de deux formes biquadratiques, Math. Ann. 11 (1877) 30-40.

J. J. Sylvester, Sur le vrai nombre des covariants élémentaires d'un système de deux formes biquadratiques binaires, Comptes Rendus 84 (1877) 1285-1289.

J. J. Sylvester & F. Franklin, Tables of the generating functions and groundforms for simultaneous binary quantics of the first four orders, taken two and two together, Amer. J. Math. 2 (1879) 293-306.

E. d'Ovidio, Sopra due covarianti simultanei di due forme binarie biquadratiche, Atti di Torino 15 (1880) 301-304.

A. Young, The irreducible concomitants of any number of binary quartics, Proc. London Math. Soc. 30 (1899) 290-307.

## V5+V1

23 invariants, 94 covariants in total.

Numbers of covariants with given degree and order:

d\o 0 1 2 3 4 5 6 7 8 9 # cum
1 - 1 - - - 1 - - - - 2 2
2 - - 1 - 1 - 1 - - - 3 5
3 - 1 - 2 - 2 - - - 1 6 11
4 2 - 2 - 3 - 1 - 1 - 9 20
5 - 3 - 4 - 1 - 2 - - 10 30
6 3 - 5 - 2 - 2 - - - 12 42
7 - 6 - 2 - 3 - - - - 11 53
8 7 - 3 - 3 - - - - - 13 66
9 - 3 - 4 - - - - - - 7 73
10 3 - 4 - - - - - - - 7 80
11 - 5 - - - - - - - - 5 85
12 6 - - - - - - - - - 6 91
13 - 1 - - - - - - - - 1 92
14 1 - - - - - - - - - 1 93
... - - - - - - - - - - - 93
18 1 - - - - - - - - - 1 94

## V5+V2

29 invariants, 92 covariants in total.

Numbers of covariants with given degree and order:

d\o 0 1 2 3 4 5 6 7 8 9 # cum
1 - - 1 - - 1 - - - - 2 2
2 1 - 1 1 - 1 1 - - - 5 7
3 1 1 1 2 1 1 1 - - 1 9 16
4 1 2 1 2 2 - 1 1 - - 10 26
5 1 3 2 1 2 1 - 1 - - 11 37
6 1 3 3 1 1 1 - - - - 10 47
7 3 3 2 1 - 1 - - - - 10 57
8 3 3 1 1 - - - - - - 8 65
9 3 2 - 1 - - - - - - 6 71
10 2 2 - - - - - - - - 4 75
11 3 2 - - - - - - - - 5 80
12 3 1 - - - - - - -   4 84
13 2 1 - - - - - -     3 87
14 1 - - - - - -       1 88
15 1 - - - - -         1 89
16 1 - - - -           1 90
17 1 - - -             1 91
18 1 - -               1 92

This system is complete, as can be checked using the algorithm in Grace & Young §141.

It is rumoured that Winter listed 94 covariants.

The same table, but with multidegrees d5.d2 given:
d\o 0 1 2 3 4 5 6 7 8 9 # cum
1     0.1     1.0         2 2
2 0.2   2.0 1.1   1.1 2.0       5 7
3 2.1 1.2 2.1 3.0/1.2 2.1 3.0 2.1     3.0 9 16
4 4.0 3.1/1.3 2.2 3.1/3.1 4.0/2.2   4.0 3.1     10 26
5 2.3 5.0/3.2/3.2 4.1/2.3 5.0 4.1/4.1 3.2   5.0     11 37
6 4.2 5.1/5.1/3.3 6.0/4.2/4.2 3.3 6.0 5.1         10 47
7 6.1/4.3/2.5 7.0/5.2/3.4 6.1/6.1 5.2   7.0         10 57
8 8.0/6.2/4.4 7.1/5.3/3.5 8.0 7.1             8 65
9 8.1/6.3/4.5 7.2/5.4   9.0             6 71
10 8.2/6.4 9.1/7.3                 4 75
11 10.1/8.3/4.7 11.0/9.2                 5 80
12 12.0/10.2/6.6 11.1                 4 84
13 12.1/8.5 13.0                 3 87
14 10.4                   1 88
15 12.3                   1 89
16 14.2                   1 90
17 16.1                   1 91
18 18.0                   1 92

### References

J. H. Grace & A. Young, The algebra of invariants, Cambridge, 1903.

E. Winter, Ueber das simultane Formensystem einer binären Form 5. Ordnung und einer binären Form 2. Ordnung, Darmstadt, Brill, 1880, 25 S.

## V5+V3

We give a possibly incomplete table.

d\o 0 1 2 3 4 5 6 7 8 9 10 # cum
1 - - - 1 - 1 - - - - - 2 2
2 - - 3 - 1 - 2 - - - - 6 8
3 - 2 - 6 - 3 - 1 - 1 - 13 21
4 6 - 6 - 6 - 2 - - - - 20 41
5 - 11 - 9 - 1 - 1 - - - 22 63
6 7 - 12 - 2 - - - - - - 21 84
7 - 20 - 1 - 1 - - - - - 22 106
8 15 - 4 1 - - - - - - - 19 125
9 - 5 - 1 - - - - - - - 6 131
10 14 - - - - - - - - - - 14 145
11 - 2 - - - - - - - - - 2 147
12 4 - - - - - - - - - - 4 151
13 - 1 - - - - - - - 1 152
14 2 - - - - - - - - 2 154
15 - - - - - - - 154
16 1 - - 1 155
17 - - - 155
18 1 1 156

There are 50 basic invariants, listed above. There may be futher basic covariants, not listed above.

## V6+V1

26 invariants, 135 covariants in total.

Numbers of covariants with given degree and order:

d\o 0 1 2 3 4 5 6 7 8 9 10 11 12 # cum
1 - 1 - - - - 1 - - - - - - 2 2
2 1 - - - 1 1 - - 1 - - - - 4 6
3 - - 1 1 1 - 1 1 1 - - - 1 7 13
4 1 1 1 1 1 1 2 1 - - 1 1 - 11 24
5 1 1 2 1 2 2 1 - 1 1 1 - - 13 37
6 2 2 1 2 2 1 2 1 1 1 - - - 15 52
7 2 1 3 2 2 2 1 1 1 - - - - 15 67
8 1 3 3 2 2 1 1 1 - - - - - 14 81
9 3 3 2 2 2 1 1 - - - - - - 14 95
10 4 2 3 2 1 1 - - - - - - - 13 108
11 2 3 2 1 1 - - - - - - - - 9 117
12 3 2 2 1 - - - - - - - - - 8 125
13 2 2 1 - - - - - - - - - - 5 130
14 2 1 - - - - - - - - - - - 3 133
15 2 - - - - - - - - - - - - 2 135

## V6+V2

27 invariants, 99 covariants in total.

Numbers of covariants with given degree and order:

d\o 0 2 4 6 8 10 12 # cum
1 - 1 - 1 - - - 2 2
2 2 - 2 1 1 - - 6 8
3 - 3 2 2 2 - 1 10 18
4 4 3 3 4 - 2 - 16 34
5 - 4 6 - 3 - - 13 47
6 5 7 - 5 - - - 17 64
7 3 1 6 - - - - 10 74
8 1 8 - - - - - 9 83
9 7 - 1 - - - - 8 91
10 1 2 - - - - - 3 94
11 2 - - - - - - 2 96
12 - 1 - - - - - 1 97
13 1 - - - - - - 1 98
14 - - - - - - - - 98
15 1 - - - - - - 1 99

This system is complete, as can be checked using the algorithm in Grace & Young §141. It was first given by A. von Gall. See also Brouwer-Popoviciu (for minimality) and Olive.

## V6+V4

60 invariants, 194 covariants in total. This table is due to Olive.

Numbers of covariants with given degree and order:

d\o 0 2 4 6 8 10 12 # cum
1 - - 1 1 - - - 2 2
2 2 1 3 1 2 - - 9 11
3 2 4 4 5 3 1 1 20 31
4 4 6 9 5 2 1 - 27 58
5 4 12 11 3 1 - - 31 89
6 9 14 6 2 - - - 31 120
7 9 17 2 - - - - 28 148
8 9 7 1 - - - - 17 165
9 8 3 1 - - - - 12 177
10 5 2 - - - - - 7 184
11 3 1 - - - - - 4 188
12 2 1 - - - - - 3 191
13 1 - - - - - - 1 192
14 1 - - - - - - 1 193
15 1 - - - - - - 1 194

### References

J. H. Grace & A. Young, The algebra of invariants, Cambridge, 1903.

A. von Gall, Ueber das simultane Formensystem einer Form 2ter und 6ter Ordnung, Jahresbericht über das Gymnasium zu Lengo, Lengo, 1874.

A. E. Brouwer & M. Popoviciu, SL2-modules of small homological dimension, Transformation Groups 16 (2011) 599-617.

Marc Olive, About Gordan's algorithm for binary forms, arXiv:1403.2283v5.