Invariants for a single form, multiple forms.

V3+V1 V3+V2 V4+V2 V4+V3 2V4 V5+V1 V5+V2 V5+V3 V6+V1 V6+V2 V6+V4

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

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

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

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

2V4

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

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

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

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

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

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

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

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.