quadric cubic quartic quintic sextic septimic octavic nonic decimic duodecimic

Invariants for several forms, multiple forms.

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

# Invariants of binary forms

A representation of the invariants of binary forms (up to nonics) in terms of bracket monomials.

The discriminant of a form of degree n is an invariant of degree 2n–2.

For even n = 2j the catalecticant (see below under quartics) is an invariant of degree j+1.

The binary quadric aX2 + 2bXY + cY2 has one invariant of degree 2, namely the discriminant.

Poincaré series: 1/(1–t2).

Numbers of basic invariants and covariants (d: degree in the coefficients, o: order in the variables):

d\o 0 2 # cum
1 - 1 1 1
2 1 - 1 2

Basic invariant in bracket form:

NameBracket monomial
I2 [1,2]^2 =
ac - b^2

## The binary cubic

The binary cubic aX3 + 3bX2Y + 3cXY2 + dY3 has one invariant of degree 4, namely the discriminant.

Poincaré series: 1/(1–t4).

Numbers of basic invariants and covariants:

d\o 0 1 2 3 # cum
1 - - - 1 1 1
2 - - 1 - 1 2
3 - - - 1 1 3
4 1 - - - 1 4

Basic invariant in bracket form:

NameBracket monomial
I4 [1,2]^2 [1,3] [2,4] [3,4]^2 =
a^2d^2 – 6abcd + 4ac^3 – 3b^2c^2 + 4b^3d

## The binary quartic

The binary quartic aX4 + 4bX3Y + 6cX2Y2 + 4dXY3 + eY4 has two invariants of degrees 2 and 3.

Poincaré series: 1/(1–t2)(1–t3).

Numbers of basic invariants and covariants:

d\o 0 2 4 6 # cum
1 - - 1 - 1 1
2 1 - 1 - 2 3
3 1 - - 1 2 5

Basic invariants in bracket form:

NameBracket monomial
I2 [1,2]^4 =
ae – 4bd + 3c^2
I3 [1,2]^2 [1,3]^2 [2,3]^2 =
ace – ad^2 – b^2e – c^3 + 2bcd

This last invariant can be written as determinant of the 3x3 matrix

 a b c b c d c d e
More generally, for forms of even degree 2j, the invariant given by ∏i<j [i,j]2, known as the catalecticant, can be written as the determinant of the (j+1)x(j+1) matrix
 a0 a1 ... aj a1 a2 ... aj+1 ... aj aj+1 ... a2j
The discriminant equals I23 – 27I32.

## The binary quintic

The binary quintic has four invariants of degrees 4, 8, 12, 18. Those of degrees 4, 8, 12 are algebraically independent, the one of degree 18 squares to an expression in the others.

Poincaré series: (1+t18)/(1–t4)(1–t8)(1–t12)

Numbers of basic invariants and covariants (Gordan (1868)):

d\o 0 1 2 3 4 5 6 7 8 9 # cum
1 - - - - - 1 - - - - 1 1
2 - - 1 - - - 1 - - - 2 3
3 - - - 1 - 1 - - - 1 3 6
4 1 - - - 1 - 1 - - - 3 9
5 - 1 - 1 - - - 1 - - 3 12
6 - - 1 - 1 - - - - - 2 14
7 - 1 - - - 1 - - - - 2 16
8 1 - 1 - - - - - - - 2 18
9 - - - 1 - - - - - - 1 19
10 - - - - - - - - - - - 19
11 - 1 - - - - - - - - 1 20
12 1 - - - - - - - - - 1 21
13 - 1 - - - - - - - - 1 22
... - - - - - - - - - - - 22
18 1 - - - - - - - - - 1 23

Basic invariants in bracket form:

NameBracket monomial
I4 [1,2]^3 [1,3]^2 [2,4]^2 [3,4]^3 =
a^2f^2 – 10abef + 4acdf + 16ace^2 – 12ad^2e + 16b^2df + 9b^2e^2 – 12bc^2f – 76bcde + 48bd^3 + 48c^3e –32c^2d^2
I8 [1,2]^3 [1,3]^2 [2,4]^2 [3,4] [3,5]^2 [4,6]^2 [5,7]^3 [6,8]^3 [7,8]^2
I12 [1,2]^2 [1,3] [1,4]^2 [2,5]^3 [3,6]^3 [3,7] [4,7] [4,8]^2 [5,8]^2 [6,8] [6,9] [7,10]^3 [9,10]^2 [9,11] [9,12] [11,12]^4
I18 [1,2] [1,3]^4 [2,4]^2 [2,5]^2 [3,5] [4,5]^2 [4,6] [6,7] [6,8]^3 [7,8] [7,9]^2 [7,10] [8,10] [9,10] [9,11]^2 [10,11]^2 [11,12] [12,13]^4 [13,14] [14,15]^3 [14,16] [15,16] [15,17] [16,17] [16,18]^2 [17,18]^3

## The binary sextic

The binary sextic has five invariants of degrees 2, 4, 6, 10, 15. Those of degrees 2, 4, 6, 10 are algebraically independent, the one of degree 15 squares to an expression in the others. The invariant of degree 10 can be taken to be the discriminant.

Poincaré series: (1+t15) / (1–t2)(1–t4)(1–t6)(1–t10)

Numbers of basic invariants and covariants (Gordan (1868)):

d\o 0 2 4 6 8 10 12 # cum
1 - - - 1 - - - 1 1
2 1 - 1 - 1 - - 3 4
3 - 1 - 1 1 - 1 4 8
4 1 - 1 1 - 1 - 4 12
5 - 1 1 - 1 - - 3 15
6 1 - - 2 - - - 3 18
7 - 1 1 - - - - 2 20
8 - 1 - - - - - 1 21
9 - - 1 - - - - 1 22
10 1 1 - - - - - 2 24
11 - - - - - - - - 24
12 - 1 - - - - - 1 25
... - - - - - - - - 25
15 1 - - - - - - 1 26

Basic invariants in bracket form:

NameBracket monomial
I2 [1,2]^6 =
ag – 6bf + 15ce – 10d^2
I4 [1,2]^4 [1,3]^2 [2,4]^2 [3,4]^4
I6 [1,2]^4 [1,3]^2 [2,4]^2 [3,5]^4 [4,6]^4 [5,6]^2
I10 [1,2]^3 [1,3]^3 [2,3]^2 [2,4] [3,5] [4,6]^4 [4,7] [5,7]^3 [5,8]^2 [6,8]^2 [7,9]^2 [8,10]^2 [9,10]^4
I15 [1,2]^3 [1,3]^3 [2,3]^2 [2,4] [3,4] [4,5] [4,6]^3 [5,6] [5,7]^4 [6,7] [6,8] [7,8] [8,9]^3 [8,10] [9,10]^3 [10,11]^2 [11,12]^4 [12,13] [12,14] [13,14]^2 [13,15]^3 [14,15]^3

## The binary septimic

The binary septimic has thirty invariants, of degrees 4, 8 (3 times), 12 (6 times), 14 (4 times), 16 (2 times), 18 (9 times), 20, 22 (2 times), 26, 30.

Poincaré series: (1 + 2t8 + 4t12 + 4t14 + 5t16 + 9t18 + 6t20 + 9t22 + 8t24 + 9t26 + 6t28 + 9t30 + 5t32 + 4t34 + 4t36 + 2t40 + t48) / (1–t4)(1–t8)(1–t12)2(1–t20)

Numbers of basic invariants (Dixmier & Lazard (1986)) and covariants (Cröni (2002)):

d\o 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 # cum
1 - - - - - - - 1 - - - - - - - - 1 1
2 - - 1 - - - 1 - - - 1 - - - - - 3 4
3 - - - 1 - 1 - 1 - 1 - 1 - - - 1 6 10
4 1 - - - 2 - 1 - 2 - 1 - - - 1 - 8 18
5 - 1 - 2 - 2 - 2 - 2 - - - 1 - - 10 28
6 - - 3 - 2 - 2 - 2 - - - 1 - - - 10 38
7 - 3 - 2 - 4 - 2 - - - 1 - - - - 12 50
8 3 - 3 - 3 - 3 - - - 1 - - - - - 13 63
9 - 3 - 5 - 2 - - - 1 - - - - - - 11 74
10 - - 4 - 4 - - - 1 - - - - - - - 9 83
11 - 5 - 3 - - - 1 - - - - - - - - 9 92
12 6 - 6 - - - 1 - - - - - - - - - 13 105
13 - 7 - 1 - 1 - - - - - - - - - - 9 114
14 4 - - - 2 - - - - - - - - - - - 6 120
15 - 3 - 1 - - - - - - - - - - - - 4 124
16 2 - 3 - - - - - - - - - - - - - 5 129
17 - 2 - - - - - - - - - - - - - - 2 131
18 9 - - - - - - - - - - - - - - - 9 140
19 - 1 - - - - - - - - - - - - - - 1 141
20 1 - - - - - - - - - - - - - - - 1 142
21 - - - - - - - - - - - - - - - - - 142
22 2 - - - - - - - - - - - - - - - 2 144
23 - 1 - - - - - - - - - - - - - - 1 145
... - - - - - - - - - - - - - - - - - 145
26 1 - - - - - - - - - - - - - - - 1 146
... - - - - - - - - - - - - - - - - - 146
30 1 - - - - - - - - - - - - - - - 1 147

Basic invariants in bracket form:

NameBracket monomial
I4 [1,2]^6 [1,3] [2,4] [3,4]^6 =
a^2h^2 – 14abgh + 18acfh + 24acg^2 – 10adeh – 60adfg + 40ae^2g + 24b^2fh + 25b^2g^2 – 60bceh – 234bcfg + 40bd^2h + 50bdeg + 360bdf^2 – 240be^2f + 360c^2eg + 81c^2f^2 – 240cd^2g – 990cdef + 600ce^3 + 600d^3f – 375d^2e^2
I8a [1,2]^6 [1,3] [2,4] [3,4]^4 [3,5]^2 [4,6]^2 [5,6]^4 [5,7] [6,8] [7,8]^6
I8b [1,2]^6 [1,3] [2,4] [3,4]^3 [3,5]^2 [3,6] [4,6]^3 [5,6]^3 [5,7] [5,8] [7,8]^6
I8c [1,2]^5 [1,3] [1,4] [2,4]^2 [3,5]^4 [3,6]^2 [4,6]^2 [4,7]^2 [5,7]^2 [5,8] [6,8]^3 [7,8]^3
I12a [1,2]^6 [1,3] [2,3] [3,4]^2 [3,5] [3,6] [3,7] [4,7] [4,8]^2 [4,9]^2 [5,9]^5 [5,10] [6,10]^4 [6,11]^2 [7,11]^5 [8,12]^5 [10,12]^2
I12b [1,2]^2 [1,3]^2 [1,4]^3 [2,4] [2,5]^4 [3,5]^2 [3,6]^3 [4,6] [4,7] [4,8] [5,8] [6,9]^3 [7,9] [7,10]^4 [7,11] [8,11]^5 [9,12]^3 [10,12]^3 [11,12]
I12c [1,2]^2 [1,3]^5 [2,3] [2,4]^3 [2,5] [3,5] [4,6]^3 [4,7] [5,8]^5 [6,9]^4 [7,9]^3 [7,10]^3 [8,10] [8,11] [10,11] [10,12]^2 [11,12]^5
I12d [1,2]^6 [1,3] [2,3] [3,4]^4 [3,5] [4,5]^2 [4,6] [5,6] [5,7]^2 [5,8] [6,8]^2 [6,9]^3 [7,9]^2 [7,10]^2 [7,11] [8,11]^4 [9,11] [9,12] [10,12]^5 [11,12]
I12e [1,2]^3 [1,3]^2 [1,4]^2 [2,4]^4 [3,5]^5 [4,5] [5,6] [6,7]^2 [6,8]^4 [7,9]^2 [7,10]^3 [8,11]^3 [9,11]^3 [9,12]^2 [10,12]^4 [11,12]
I12f [1,2]^4 [1,3] [1,4]^2 [2,5] [2,6]^2 [3,6]^3 [3,7]^3 [4,7] [4,8]^4 [5,8]^2 [5,9]^3 [5,10] [6,10]^2 [7,10]^2 [7,11] [8,11] [9,11]^2 [9,12]^2 [10,12]^2 [11,12]^3
I14a [1,2]^2 [1,3]^5 [2,3] [2,4]^4 [3,5] [4,5]^2 [4,6] [5,6]^4 [6,7]^2 [7,8]^4 [7,9] [8,9]^2 [8,10] [9,11]^3 [9,12] [10,12]^3 [10,13]^3 [11,13]^2 [11,14]^2 [12,14]^3 [13,14]^2
I14b [1,2]^6 [1,3] [2,4] [3,4] [3,5]^5 [4,5] [4,6]^4 [5,6] [6,7]^2 [7,8]^3 [7,9]^2 [8,9] [8,10]^2 [8,11] [9,11]^4 [10,11] [10,12]^4 [11,12] [12,13] [12,14] [13,14]^6
I14c [1,2]^6 [1,3] [2,3] [3,4]^4 [3,5] [4,5] [4,6]^2 [5,6]^4 [5,7] [6,7] [7,8]^3 [7,9]^2 [8,9]^3 [8,10] [9,10] [9,11] [10,11]^2 [10,12]^2 [10,13] [11,13]^4 [12,14]^5 [13,14]^2
I14d [1,2]^3 [1,3]^2 [1,4]^2 [2,4] [2,5]^2 [2,6] [3,6]^5 [4,6] [4,7]^2 [4,8] [5,8]^5 [7,9] [7,10]^2 [7,11]^2 [8,11] [9,11]^4 [9,12]^2 [10,13]^5 [12,14]^5 [13,14]^2
I16a [1,2]^4 [1,3] [1,4] [1,5] [2,5]^3 [3,5]^2 [3,6]^2 [3,7]^2 [4,7]^3 [4,8]^2 [4,9] [5,9] [6,10]^4 [6,11] [7,11]^2 [8,11] [8,12]^4 [9,12] [9,13]^4 [10,14]^2 [10,15] [11,15]^3 [12,15]^2 [13,15] [13,16]^2 [14,16]^5
I16b [1,2] [1,3]^6 [2,3] [2,4]^4 [2,5] [4,5]^3 [5,6]^3 [6,7]^3 [6,8] [7,8]^2 [7,9] [7,10] [8,10]^4 [9,10]^2 [9,11] [9,12]^3 [11,13]^6 [12,14]^2 [12,15]^2 [13,15] [14,15] [14,16]^4 [15,16]^3
I18a [1,2]^5 [1,3]^2 [2,4]^2 [3,5]^4 [3,6] [4,6]^5 [5,7]^3 [6,7] [7,8]^2 [7,9] [8,9]^2 [8,10]^3 [9,11]^4 [10,11]^2 [10,12]^2 [11,13] [12,14]^5 [13,15]^4 [13,16]^2 [14,16] [14,17] [15,17]^3 [16,18]^4 [17,18]^3
I18b [1,2]^3 [1,3]^3 [1,4] [2,4]^3 [2,5] [3,5]^4 [4,6]^3 [5,7]^2 [6,8]^2 [6,9]^2 [7,9]^3 [7,10]^2 [8,10]^2 [8,11]^3 [9,11] [9,12] [10,13]^3 [11,13]^2 [11,14] [12,14]^4 [12,15]^2 [13,15]^2 [14,15] [14,16] [15,17]^2 [16,17]^2 [16,18]^4 [17,18]^3
I18c [1,2]^6 [1,3] [2,4] [3,4]^2 [3,5]^4 [4,6]^4 [5,6]^2 [5,7] [6,7] [7,8]^5 [8,9]^2 [9,10]^2 [9,11]^3 [10,11]^3 [10,12] [10,13] [11,13] [12,13]^4 [12,14]^2 [13,14] [14,15]^4 [15,16]^2 [15,17] [16,17]^2 [16,18]^3 [17,18]^4
I18d [1,2]^4 [1,3] [1,4]^2 [2,4]^3 [3,4]^2 [3,5]^3 [3,6] [5,7]^2 [5,8]^2 [6,8]^2 [6,9]^3 [6,10] [7,10]^3 [7,11]^2 [8,11]^3 [9,12]^4 [10,13]^3 [11,13]^2 [12,13] [12,14]^2 [13,15] [14,15] [14,16]^4 [15,17]^4 [15,18] [16,18]^3 [17,18]^3
I18e [1,2]^6 [1,3] [2,3] [3,4]^2 [3,5]^2 [3,6] [4,6]^4 [4,7] [5,7]^3 [5,8]^2 [6,8]^2 [7,8]^2 [7,9] [8,9] [9,10]^2 [9,11]^3 [10,11]^4 [10,12] [12,13]^5 [12,14] [13,14] [13,15] [14,15]^4 [14,16] [15,16] [15,17] [16,17]^2 [16,18]^3 [17,18]^4
I18f [1,2] [1,3]^5 [1,4] [2,4]^3 [2,5]^3 [3,5]^2 [4,5] [4,6]^2 [5,7] [6,7]^2 [6,8]^3 [7,8]^3 [7,9] [8,10] [9,10]^4 [9,11] [9,12] [10,13]^2 [11,13]^2 [11,14]^4 [12,14]^2 [12,15]^2 [12,16]^2 [13,16]^3 [14,16] [15,17]^3 [15,18]^2 [16,18] [17,18]^4
I18g [1,2]^2 [1,3]^4 [1,4] [2,4]^2 [2,5]^3 [3,5]^3 [4,5] [4,6]^3 [6,7]^2 [6,8]^2 [7,8]^2 [7,9]^3 [8,9] [8,10]^2 [9,10]^3 [10,11]^2 [11,12]^3 [11,13]^2 [12,13]^2 [12,14] [12,15] [13,15]^3 [14,16]^6 [15,17]^2 [15,18] [16,18] [17,18]^5
I18h [1,2]^3 [1,3]^2 [1,4]^2 [2,4] [2,5]^2 [2,6] [3,6] [3,7]^4 [4,8]^3 [4,9] [5,10]^5 [6,10]^2 [6,11]^2 [6,12] [7,13]^2 [7,14] [8,14]^4 [9,15]^6 [11,15] [11,16]^4 [12,16] [12,17]^5 [13,17]^2 [13,18]^3 [14,18]^2 [16,18]^2
I18i [1,2]^4 [1,3]^3 [2,3] [2,4]^2 [3,5]^3 [4,5]^3 [4,6] [4,7] [5,7] [6,7]^4 [6,8]^2 [7,9] [8,10]^2 [8,11]^2 [8,12] [9,12] [9,13]^5 [10,14]^2 [10,15]^3 [11,16]^5 [12,16]^2 [12,17]^3 [13,17]^2 [14,17]^2 [14,18]^3 [15,18]^4
I20 [1,2]^3 [1,3]^3 [1,4] [2,4]^4 [3,5] [3,6]^3 [4,7]^2 [5,7] [5,8]^3 [5,9]^2 [6,9]^3 [6,10] [7,10]^4 [8,10]^2 [8,11] [8,12] [9,13] [9,14] [11,14]^4 [11,15]^2 [12,15]^2 [12,16] [12,17]^3 [13,17]^4 [13,18]^2 [14,18] [14,19] [15,19]^3 [16,19]^3 [16,20]^3 [18,20]^4
I22a [1,2]^3 [1,3]^4 [2,3]^2 [2,4]^2 [3,5] [4,5]^2 [4,6]^3 [5,6]^3 [5,7] [6,8] [7,8]^3 [7,9]^3 [8,9]^2 [8,10] [9,10]^2 [10,11] [10,12]^2 [10,13] [11,14]^4 [11,15]^2 [12,15]^2 [12,16]^3 [13,16]^2 [13,17]^2 [13,18]^2 [14,18]^3 [15,18] [15,19] [15,20] [16,20]^2 [17,20]^2 [17,21]^3 [18,21] [19,21]^2 [19,22]^4 [20,22]^2 [21,22]
I22b [1,2]^3 [1,3]^4 [2,3] [2,4]^3 [3,4] [3,5] [4,6]^2 [4,7] [5,7]^2 [5,8]^2 [5,9]^2 [6,9]^4 [6,10] [7,10]^4 [8,11]^2 [8,12]^2 [8,13] [9,13] [10,14]^2 [11,14] [11,15]^4 [12,16]^4 [12,17] [13,17] [13,18]^4 [14,19]^4 [15,19] [15,20]^2 [16,20]^3 [17,20]^2 [17,21]^3 [18,21] [18,22]^2 [19,22]^2 [21,22]^3
I26 [1,2]^2 [1,3]^2 [1,4]^3 [2,4]^3 [2,5]^2 [3,5]^2 [3,6]^3 [4,6] [5,7]^2 [5,8] [6,8]^3 [7,8]^2 [7,9]^2 [7,10] [8,10] [9,10] [9,11]^4 [10,11] [10,12]^2 [10,13] [11,13]^2 [12,13]^2 [12,14]^3 [13,14]^2 [14,15]^2 [15,16]^4 [15,17] [16,17]^2 [16,18] [17,18]^2 [17,19]^2 [18,19] [18,20]^3 [19,21]^2 [19,22]^2 [20,22]^4 [21,22] [21,23]^2 [21,24]^2 [23,24]^4 [23,25] [24,26] [25,26]^6
I30 [1,2]^2 [1,3]^2 [1,4]^3 [2,4]^4 [2,5] [3,5]^2 [3,6]^2 [3,7] [5,7]^4 [6,8]^3 [6,9]^2 [7,9]^2 [8,9]^2 [8,10] [8,11] [9,11] [10,11]^2 [10,12]^4 [11,13]^3 [12,13] [12,14]^2 [13,15]^2 [13,16] [14,16]^4 [14,17] [15,17]^3 [15,18] [15,19] [16,19]^2 [17,20]^3 [18,21]^2 [18,22]^4 [19,22]^2 [19,23]^2 [20,23]^2 [20,24]^2 [21,24] [21,25]^4 [22,25] [23,25]^2 [23,26] [24,26]^3 [24,27] [26,27]^3 [27,28] [27,29]^2 [28,29]^2 [28,30]^4 [29,30]^3

## The binary octavic

The binary octavic has nine invariants of degrees 2, 3, 4, 5, 6, 7, 8, 9, 10.

Poincaré series: (1+t8+t9+t10+t18) / (1–t2)(1–t3)(1–t4)(1–t5)(1–t6)(1–t7)

Numbers of basic invariants and covariants:

d\o 0 2 4 6 8 10 12 14 16 18 # cum
1 - - - - 1 - - - - - 1 1
2 1 - 1 - 1 - 1 - - - 4 5
3 1 - 1 1 1 1 1 1 - 1 8 13
4 1 - 2 1 1 2 1 1 - 1 10 23
5 1 1 2 2 1 3 - 1 - - 11 34
6 1 1 2 3 1 1 - - - - 9 43
7 1 2 2 3 - - - - - - 8 51
8 1 2 2 2 - - - - - - 7 58
9 1 3 1 - - - - - - - 5 63
10 1 2 - - - - - - - - 3 66
11 - 2 - - - - - - - - 2 68
12 - 1 - - - - - - - - 1 69

(See Sylvester & Franklin (1879), von Gall (1880), Shioda (1967), Cröni (2002), Bedratyuk (2006).)

Basic invariants in bracket form:

NameBracket monomial
I2 [1,2]^8 =
ai – 8bh + 28cg – 56df + 35e^2
I3 [1,2]^4 [1,3]^4 [2,3]^4 =
aei – 4afh + 3ag^2 – 4bdi + 12beh – 8bfg + 3c^2i – 8cdh – 22ceg + 24cf^2 + 24d^2g – 36def + 15e^3
I4 [1,2]^4 [1,3]^4 [2,4]^4 [3,4]^4
I5 [1,2]^4 [2,3]^4 [3,4]^4 [4,5]^4 [1,5]^4
I6 [1,2]^4 [2,3]^4 [3,4]^2 [3,5]^2 [4,5]^4 [4,6]^2 [5,6]^2 [1,6]^4
I7 [1,2]^2 [1,3]^2 [2,3]^4 [2,4]^2 [3,5]^2 [4,5]^4 [4,6]^2 [5,6]^2 [6,7]^4 [1,7]^4
I8 [1,2]^2 [1,3]^2 [2,3]^4 [2,4]^2 [3,5]^2 [4,5]^4 [4,6]^2 [5,6]^2 [6,7]^4 [7,8]^4 [1,8]^4
I9 [1,2]^2 [1,3]^2 [2,3]^4 [2,4]^2 [3,5]^2 [4,5]^4 [4,6]^2 [5,6]^2 [6,7]^4 [7,8]^2 [7,9]^2 [8,9]^4 [1,8]^2 [1,9]^2
I10 [1,2]^2 [1,3]^2 [2,3]^4 [2,4]^2 [3,5]^2 [4,5]^4 [4,6]^2 [5,6]^2 [6,7]^4 [7,8]^2 [7,9]^2 [8,9]^4 [8,10]^2 [9,10]^2 [1,10]^4

## The binary nonic

Dixmier (LNM 1146) shows that there exists a homogeneous system of parameters (hsop) with degrees 4, 8, 10, 12, 12, 14, 16. An explicit hsop was given by aeb & M. Popoviciu. We also show that there exist hsops with degrees 4, 4, 10, 12, 14, 16, 24 and with degrees 4, 4, 8, 12, 14, 16, 30 and with degrees 4, 4, 8, 10, 12, 16, 42 and with degrees 4, 4, 8, 10, 12, 14, 48.

Cröni gives an incomplete list of degrees of basic invariants, missing only the largest degree. Here we give the full list of degrees and the invariants themselves. The degrees are 4 (2 times), 8 (5 times), 10 (5 times), 12 (14 times), 14 (17 times), 16 (21 times), 18 (25 times), 20 (2 times), 22 (once) for a total of 92 basic invariants. This settles a 130-year-old problem.

Poincaré series:
(1 + t4 + 5t8 + 4t10 + 17t12 + 20t14 + 47t16 + 61t18 + 97t20 + 120t22 + 165t24 + 189t26 + 223t28 + 241t30 + 254t32 + 254t34 + 241t36 + 223t38 + 189t40 + 165t42 + 120t44 + 97t46 + 61t48 + 47t50 + 20t52 + 17t54 + 4t56 + 5t58 + t62 + t66) /
(1–t4)(1–t8)(1–t10)(1–t12)2(1–t14)(1–t16)

Numbers of basic invariants and covariants (the latter possibly incomplete):

d\o 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 # cum
1 - - - - - - - - - 1 - - - - - - - - - - - - - 1 1
2 - - 1 - - - 1 - - - 1 - - - 1 - - - - - - - - 4 5
3 - - - 1 - 1 - 1 - 2 - 1 - 1 - 1 - 1 - - - 1 - 10 15
4 2 - - - 2 - 2 - 3 - 2 - 2 - 2 - 1 - 1 - - - 1 18 33
5 - 1 - 3 - 4 - 4 - 3 - 4 - 2 - 3 - - - 1 - - - 25 58
6 - - 4 - 4 - 6 - 6 - 3 - 4 - - - 1 - - - - - - 28 86
7 - 4 - 7 - 8 - 7 - 6 - 1 - 1 - - - - - - - - - 34 120
8 5 - 8 - 10 - 10 - 4 - 2 - - - - - - - - - - - - 39 159
9 - 9 - 14 - 10 - 7 - 1 - - - - - - - - - - - - - 41 200
10 5 - 15 - 15 - 3 - 1 - - - - - - - - - - - - - - 39 239
11 - 17 - 16 - 7 - 1 - - - - - - - - - - - - - - - 41 280
12 14 - 23 - 4 - 1 - - - - - - - - - - - - - - - - 42 322
13 - 25 - 10 - 1 - - - - - - - - - - - - - - - - - 36 358
14 17 - 13 - 1 - - - - - - - - - - - - - - - - - - 31 389
15 - 26 - 1 - - - - - - - - - - - - - - - - - - - 27 416
16 21 - 3 - - - - - - - - - - - - - - - - - - - - 24 440
17 - 7 - - - - - - - - - - - - - - - - - - - - - 7 447
18 25 - - - - - - - - - - - - - - - - - - - - - - 25 472
19 - 1 - - - - - - - - - - - - - - - - - - - - - 1 473
20 2 - - - - - - - - - - - - - - - - - - - - - - 2 475
21 - - - - - - - - - - - - - - - - - - - - - - - - 475
22 1 - - - - - - - - - - - - - - - - - - - - - - 1 476

(Partial results computed by Holger Cröni, Tom Hagedorn, aeb. Any further basic covariant has degree at least 25.)

Basic invariants in bracket form:

NameBracket monomial
I4a [1,2]^8 [1,3] [2,4] [3,4]^8 =
a^2j^2 – 18abij + 40achj + 32aci^2 – 56adgj – 112adhi + 28aefj + 224aegi – 140af^2i + 32b^2hj + 49b^2i^2 – 112bcgj – 536bchi + 224bdfj + 392bdgi + 896bdh^2 – 140be^2j – 196befi – 1792begh + 1120bf^2h + 896c^2gi + 400c^2h^2 – 1792cdfi – 4256cdgh + 1120ce^2i + 560cefh + 6272ceg^2 – 3920cf^2g + 6272d^2fh + 784d^2g^2 – 3920de^2h – 13328defg + 7840df^3 + 7840e^3g – 4704e^2f^2
I4b [1,2]^5 [1,3]^4 [2,4]^4 [3,4]^5
I8a [1,2]^5 [1,3]^4 [2,4]^4 [3,5]^5 [4,6]^5 [5,7]^4 [6,8]^4 [7,8]^5
I8b [1,2]^3 [1,3]^6 [2,4]^6 [3,5]^3 [4,6]^3 [5,7]^6 [6,8]^6 [7,8]^3
I8c [1,2]^6 [1,3]^3 [2,3]^2 [2,4] [3,4]^2 [3,5]^2 [4,5] [4,6]^5 [5,7]^5 [5,8] [6,8]^4 [7,8]^4
I8d [1,2] [1,3]^8 [2,3] [2,4]^3 [2,5]^4 [4,5] [4,6]^5 [5,6] [5,7]^3 [6,8]^3 [7,8]^6
I8e [1,2]^5 [1,3]^4 [2,3]^3 [2,4] [3,4]^2 [4,5]^2 [4,6]^2 [4,7]^2 [5,7]^6 [5,8] [6,8]^7 [7,8]
I10a [1,2]^6 [1,3] [1,4]^2 [2,4]^3 [3,4] [3,5]^2 [3,6]^4 [3,7] [4,7] [4,8]^2 [5,8]^7 [6,9]^5 [7,9] [7,10]^6 [9,10]^3
I10b [1,2]^4 [1,3]^5 [2,3]^2 [2,4] [2,5]^2 [3,5]^2 [4,6]^5 [4,7]^3 [5,7] [5,8]^4 [6,9]^4 [7,9]^3 [7,10]^2 [8,10]^5 [9,10]^2
I10c [1,2]^6 [1,3] [1,4]^2 [2,4]^3 [3,4]^3 [3,5] [3,6] [3,7]^3 [4,7] [5,7]^3 [5,8]^4 [5,9] [6,9]^7 [6,10] [7,10]^2 [8,10]^5 [9,10]
I10d [1,2]^4 [1,3]^3 [1,4]^2 [2,4]^5 [3,4]^2 [3,5] [3,6]^3 [5,6]^3 [5,7]^4 [5,8] [6,9]^3 [7,9]^5 [8,10]^8 [9,10]
I10e [1,2]^6 [1,3]^3 [2,3] [2,4]^2 [3,5]^2 [3,6]^3 [4,6]^2 [4,7]^5 [5,7]^4 [5,8]^3 [6,8]^2 [6,9]^2 [8,9] [8,10]^3 [9,10]^6
I12a [1,2]^8 [1,3] [2,4] [3,5]^4 [3,6]^4 [4,6]^5 [4,7]^3 [5,7]^3 [5,8]^2 [7,8] [7,9]^2 [8,9] [8,10]^3 [8,11]^2 [9,11]^5 [9,12] [10,12]^6 [11,12]^2
I12b [1,2]^8 [1,3] [2,4] [3,4]^2 [3,5]^4 [3,6]^2 [4,6]^5 [4,7] [5,7]^5 [6,8]^2 [7,9]^3 [8,9]^5 [8,10]^2 [9,10] [10,11]^3 [10,12]^3 [11,12]^6
I12c [1,2] [1,3]^5 [1,4]^3 [2,4]^6 [2,5]^2 [3,5]^2 [3,6] [3,7] [5,7]^5 [6,8] [6,9]^3 [6,10]^4 [7,10]^2 [7,11] [8,11]^8 [9,12]^6 [10,12]^3
I12d [1,2] [1,3]^3 [1,4]^5 [2,4] [2,5]^4 [2,6]^3 [3,6] [3,7]^5 [4,7]^2 [4,8] [5,8] [5,9]^3 [5,10] [6,10]^5 [7,10]^2 [8,10] [8,11]^6 [9,12]^6 [11,12]^3
I12e [1,2]^3 [1,3]^6 [2,3] [2,4]^5 [3,4] [3,5] [4,5]^2 [4,6] [5,6]^2 [5,7]^4 [6,7]^4 [6,8]^2 [7,9] [8,9]^5 [8,10]^2 [9,11]^3 [10,11]^2 [10,12]^5 [11,12]^4
I12f [1,2]^4 [1,3]^2 [1,4]^3 [2,4] [2,5]^2 [2,6]^2 [3,6]^4 [3,7] [3,8]^2 [4,8]^4 [4,9] [5,9]^6 [5,10] [6,10]^3 [7,10] [7,11]^7 [8,11] [8,12]^2 [9,12]^2 [10,12]^4 [11,12]
I12g [1,2]^3 [1,3]^4 [1,4]^2 [2,5]^6 [3,5]^2 [3,6]^2 [3,7] [4,7]^2 [4,8]^3 [4,9]^2 [5,9] [6,9]^5 [6,10]^2 [7,10]^5 [7,11] [8,11]^4 [8,12]^2 [9,12] [10,12]^2 [11,12]^4
I12h [1,2] [1,3]^6 [1,4]^2 [2,4]^3 [2,5]^2 [2,6]^3 [3,6]^3 [4,6]^3 [4,7] [5,7] [5,8] [5,9]^5 [7,9]^4 [7,10]^3 [8,10]^6 [8,11] [8,12] [11,12]^8
I12i [1,2]^8 [1,3] [2,3] [3,4]^5 [3,5]^2 [4,6]^3 [4,7] [5,7]^5 [5,8] [5,9] [6,9]^6 [7,10]^3 [8,10]^4 [8,11]^4 [9,12]^2 [10,12]^2 [11,12]^5
I12j [1,2] [1,3] [1,4]^4 [1,5]^3 [2,5]^3 [2,6]^4 [2,7] [3,7]^3 [3,8]^5 [4,9]^5 [5,9] [5,10]^2 [6,10]^5 [7,10]^2 [7,11]^3 [8,11]^2 [8,12]^2 [9,12]^3 [11,12]^4
I12k [1,2]^2 [1,3]^3 [1,4]^4 [2,4]^5 [2,5] [2,6] [3,6]^4 [3,7]^2 [5,7]^3 [5,8]^4 [5,9] [6,9]^4 [7,9]^3 [7,10] [8,10]^3 [8,11]^2 [9,11] [10,11] [10,12]^4 [11,12]^5
I12l [1,2]^6 [1,3]^3 [2,3]^3 [3,4] [3,5]^2 [4,5]^5 [4,6]^3 [5,6]^2 [6,7]^4 [7,8]^5 [8,9]^2 [8,10]^2 [9,10]^2 [9,11]^5 [10,12]^5 [11,12]^4
I12m [1,2]^8 [1,3] [2,3] [3,4] [3,5] [3,6]^5 [4,6]^3 [4,7]^5 [5,7]^2 [5,8]^3 [5,9]^3 [6,9] [7,9]^2 [8,10]^6 [9,11]^3 [10,12]^3 [11,12]^6
I12n [1,2]^5 [1,3]^4 [2,3]^2 [2,4]^2 [3,4]^2 [3,5] [4,6]^5 [5,7]^5 [5,8]^2 [5,9] [6,9]^3 [6,10] [7,10]^3 [7,11] [8,11]^7 [9,11] [9,12]^4 [10,12]^5
I14a [1,2]^3 [1,3]^2 [1,4] [1,5] [1,6]^2 [2,7]^5 [2,8] [3,8] [3,9]^4 [3,10]^2 [4,10]^7 [4,11] [5,11]^6 [5,12]^2 [6,12]^7 [7,13]^4 [8,13]^5 [8,14]^2 [9,14]^5 [11,14]^2
I14b [1,2]^6 [1,3]^3 [2,3] [2,4]^2 [3,4]^3 [3,5]^2 [4,5]^3 [4,6] [5,6]^4 [6,7]^4 [7,8]^3 [7,9]^2 [8,9]^6 [9,10] [10,11]^4 [10,12]^4 [11,13]^5 [12,14]^5 [13,14]^4
I14c [1,2]^2 [1,3]^5 [1,4]^2 [2,4]^3 [2,5] [2,6]^3 [3,6]^2 [3,7]^2 [4,8]^4 [5,8] [5,9]^3 [5,10]^4 [6,10]^2 [6,11]^2 [7,12]^7 [8,12]^2 [8,13]^2 [9,13]^6 [10,13] [10,14]^2 [11,14]^7
I14d [1,2]^6 [1,3]^3 [2,4]^3 [3,4] [3,5] [3,6] [3,7]^3 [4,7]^5 [5,7] [5,8]^2 [5,9]^5 [6,9] [6,10]^4 [6,11]^3 [8,11]^6 [8,12] [9,12]^3 [10,12]^3 [10,13]^2 [12,14]^2 [13,14]^7
I14e [1,2]^8 [1,3] [2,4] [3,4]^4 [3,5]^4 [4,5]^2 [4,6] [4,7] [5,8]^3 [6,8] [6,9]^7 [7,9]^2 [7,10]^4 [7,11]^2 [8,11]^4 [8,12] [10,13]^5 [11,13]^3 [12,14]^8 [13,14]
I14f [1,2]^2 [1,3]^4 [1,4]^3 [2,4]^2 [2,5]^3 [2,6]^2 [3,7] [3,8]^4 [4,8]^2 [4,9]^2 [5,9]^5 [5,10] [6,10]^7 [7,11]^8 [8,12]^3 [9,12]^2 [10,13] [11,13] [12,13] [12,14]^3 [13,14]^6
I14g [1,2]^2 [1,3]^4 [1,4]^3 [2,4]^2 [2,5]^5 [3,5]^2 [3,6]^3 [4,6]^4 [5,6] [5,7] [6,7] [7,8]^3 [7,9]^4 [8,10]^6 [9,11]^5 [10,12]^3 [11,12]^4 [12,13] [12,14] [13,14]^8
I14h [1,2]^6 [1,3]^2 [1,4] [2,4]^3 [3,4]^3 [3,5]^4 [4,5]^2 [5,6]^3 [6,7]^3 [6,8]^2 [6,9] [7,9]^6 [8,9]^2 [8,10]^5 [10,11]^2 [10,12]^2 [11,12]^4 [11,13]^3 [12,14]^3 [13,14]^6
I14i [1,2]^3 [1,3]^6 [2,3]^3 [2,4] [2,5]^2 [4,5]^5 [4,6] [4,7]^2 [5,7]^2 [6,8]^5 [6,9]^3 [7,10]^4 [7,11] [8,11]^2 [8,12]^2 [9,12]^3 [9,13]^3 [10,13]^5 [11,13] [11,14]^5 [12,14]^4
I14j [1,2]^7 [1,3]^2 [2,4]^2 [3,5]^7 [4,5] [4,6]^6 [5,6] [6,7]^2 [7,8]^2 [7,9]^5 [8,9] [8,10] [8,11] [8,12]^4 [9,12]^3 [10,13]^8 [11,13] [11,14]^7 [12,14]^2
I14k [1,2]^5 [1,3]^4 [2,3]^3 [2,4] [3,4]^2 [4,5]^6 [5,6]^2 [5,7] [6,7]^2 [6,8]^5 [7,8]^2 [7,9]^4 [8,9]^2 [9,10]^2 [9,11] [10,11]^2 [10,12] [10,13]^4 [11,13]^5 [11,14] [12,14]^8
I14l [1,2]^6 [1,3]^3 [2,3]^2 [2,4] [3,4]^2 [3,5]^2 [4,5]^2 [4,6]^4 [5,6]^4 [5,7] [6,7] [7,8]^4 [7,9]^3 [8,9]^3 [8,10]^2 [9,10]^3 [10,11]^2 [10,12]^2 [11,12]^6 [11,13] [12,14] [13,14]^8
I14m [1,2]^3 [1,3]^2 [1,4]^4 [2,5]^6 [3,5]^3 [3,6]^4 [4,7]^5 [6,7] [6,8]^4 [7,8] [7,9]^2 [8,9]^4 [9,10]^2 [9,11] [10,11] [10,12]^6 [11,13]^5 [11,14]^2 [12,14]^3 [13,14]^4
I14n [1,2]^5 [1,3]^4 [2,4]^4 [3,5]^5 [4,5] [4,6]^4 [5,6] [5,7]^2 [6,7] [6,8]^3 [7,8]^4 [7,9]^2 [8,10]^2 [9,10] [9,11]^6 [10,11] [10,12]^5 [11,13]^2 [12,13] [12,14]^3 [13,14]^6
I14o [1,2]^8 [1,3] [2,4] [3,4]^2 [3,5] [3,6]^5 [4,7]^4 [4,8] [4,9] [5,9]^5 [5,10]^3 [6,10]^4 [7,10] [7,11] [7,12]^3 [8,12]^4 [8,13]^4 [9,13]^3 [10,13] [11,13] [11,14]^7 [12,14]^2
I14p [1,2]^2 [1,3]^4 [1,4]^2 [1,5] [2,5]^2 [2,6]^3 [2,7]^2 [3,7]^5 [4,7] [4,8]^3 [4,9]^3 [5,9]^6 [6,10]^6 [7,10] [8,10]^2 [8,11]^2 [8,12]^2 [11,12]^5 [11,13]^2 [12,14]^2 [13,14]^7
I14q [1,2]^5 [1,3] [1,4]^3 [2,4]^4 [3,4]^2 [3,5] [3,6] [3,7]^4 [5,7]^2 [5,8]^3 [5,9]^3 [6,9]^5 [6,10]^3 [7,11] [7,12]^2 [8,12]^6 [9,13] [10,13]^6 [11,13] [11,14]^7 [12,14] [13,14]
I16a [1,2]^6 [1,3]^2 [1,4] [2,4]^3 [3,4]^2 [3,5]^5 [4,6]^3 [5,7]^3 [5,8] [6,9]^6 [7,10]^6 [8,10]^2 [8,11]^6 [9,12]^3 [10,13] [11,14]^3 [12,14] [12,15]^5 [13,15]^4 [13,16]^4 [14,16]^5
I16b [1,2]^7 [1,3]^2 [2,3] [2,4] [3,4]^4 [3,5]^2 [4,5] [4,6]^3 [5,6]^4 [5,7] [5,8] [6,8]^2 [7,8]^5 [7,9]^3 [8,10] [9,10] [9,11]^5 [10,11]^2 [10,12]^3 [10,13]^2 [11,14]^2 [12,14] [12,15]^5 [13,15]^4 [13,16]^3 [14,16]^6
I16c [1,2] [1,3]^2 [1,4]^5 [1,5] [2,5]^6 [2,6]^2 [3,6]^6 [3,7] [4,7] [4,8]^3 [5,8]^2 [6,8] [7,8]^3 [7,9]^4 [9,10] [9,11] [9,12]^3 [10,12]^3 [10,13]^5 [11,14]^8 [12,14] [12,15]^2 [13,15] [13,16]^3 [15,16]^6
I16d [1,2]^4 [1,3]^5 [2,3]^3 [2,4]^2 [3,5] [4,5]^2 [4,6]^5 [5,6]^3 [5,7]^3 [6,8] [7,8]^6 [8,9] [8,10] [9,10]^5 [9,11] [9,12]^2 [10,12] [10,13] [10,14] [11,14]^6 [11,15]^2 [12,15]^6 [13,15] [13,16]^7 [14,16]^2
I16e [1,2] [1,3]^7 [1,4] [2,4]^3 [2,5]^3 [2,6]^2 [3,6]^2 [4,6]^4 [4,7] [5,7]^6 [6,8] [7,8] [7,9] [8,10]^4 [8,11]^3 [9,11]^6 [9,12] [9,13] [10,14]^5 [12,14] [12,15]^7 [13,15]^2 [13,16]^6 [14,16]^3
I16f [1,2]^3 [1,3]^6 [2,4]^3 [2,5]^3 [3,5]^3 [4,6] [4,7]^3 [4,8]^2 [5,8] [5,9]^2 [6,9] [6,10]^6 [6,11] [7,11]^6 [8,11] [8,12] [8,13]^4 [9,13]^4 [9,14]^2 [10,14]^3 [11,14] [12,14]^3 [12,15]^3 [12,16]^2 [13,16] [15,16]^6
I16g [1,2]^5 [1,3]^4 [2,3]^3 [2,4] [3,4] [3,5] [4,5]^5 [4,6]^2 [5,7]^3 [6,7] [6,8]^4 [6,9]^2 [7,9]^5 [8,10]^2 [8,11]^3 [9,11] [9,12] [10,12]^4 [10,13]^3 [11,13]^4 [11,14] [12,15]^4 [13,15]^2 [14,15] [14,16]^7 [15,16]^2
I16h [1,2] [1,3]^7 [1,4] [2,4]^6 [2,5]^2 [3,5]^2 [4,5]^2 [5,6] [5,7]^2 [6,7]^3 [6,8]^5 [7,8]^2 [7,9]^2 [8,10]^2 [9,10]^3 [9,11]^4 [10,11]^2 [10,12]^2 [11,12] [11,13] [11,14] [12,14]^2 [12,15]^4 [13,15]^5 [13,16]^3 [14,16]^6
I16i [1,2]^6 [1,3]^3 [2,4]^2 [2,5] [3,5]^2 [3,6]^4 [4,6] [4,7]^6 [5,8]^3 [5,9]^3 [6,9]^2 [6,10]^2 [7,10]^3 [8,10]^3 [8,11]^3 [9,11] [9,12]^3 [10,12] [11,12] [11,13]^4 [12,13]^2 [12,14]^2 [13,14]^2 [13,15] [14,15]^2 [14,16]^3 [15,16]^6
I16j [1,2]^8 [1,3] [2,4] [3,4] [3,5]^2 [3,6]^4 [3,7] [4,7]^7 [5,7] [5,8]^6 [6,8]^2 [6,9]^3 [8,9] [9,10] [9,11]^3 [9,12] [10,12]^5 [10,13]^2 [10,14] [11,14]^3 [11,15]^3 [12,15]^3 [13,15]^3 [13,16]^4 [14,16]^5
I16k [1,2]^8 [1,3] [2,4] [3,4]^3 [3,5]^4 [3,6] [4,7]^5 [5,8]^5 [6,8]^4 [6,9]^3 [6,10] [7,10] [7,11] [7,12] [7,13] [9,13]^5 [9,14] [10,14]^7 [11,14] [11,15]^7 [12,15]^2 [12,16]^6 [13,16]^3
I16l [1,2]^5 [1,3]^2 [1,4]^2 [2,4]^3 [2,5] [3,6]^7 [4,7]^4 [5,7]^5 [5,8]^2 [5,9] [6,9] [6,10] [8,10]^2 [8,11]^3 [8,12]^2 [9,12]^2 [9,13]^5 [10,13]^2 [10,14]^4 [11,14]^3 [11,15]^3 [12,15]^3 [12,16]^2 [13,16]^2 [14,16]^2 [15,16]^3
I16m [1,2]^3 [1,3]^6 [2,3]^2 [2,4] [2,5]^3 [3,5] [4,5]^2 [4,6]^6 [5,6] [5,7]^2 [6,7]^2 [7,8]^5 [8,9]^3 [8,10] [9,10]^2 [9,11]^4 [10,12]^4 [10,13]^2 [11,13]^3 [11,14]^2 [12,14]^5 [13,15]^3 [13,16] [14,16]^2 [15,16]^6
I16n [1,2]^6 [1,3]^3 [2,3] [2,4]^2 [3,5]^5 [4,5]^4 [4,6] [4,7]^2 [6,7]^7 [6,8] [8,9]^8 [9,10] [10,11]^6 [10,12]^2 [11,13]^3 [12,13]^5 [12,14]^2 [13,15] [14,15]^3 [14,16]^4 [15,16]^5
I16o [1,2]^2 [1,3]^4 [1,4]^2 [1,5] [2,5]^7 [3,5] [3,6] [3,7]^3 [4,7]^5 [4,8]^2 [6,8]^3 [6,9]^4 [6,10] [7,10] [8,10]^4 [9,11]^2 [9,12] [9,13]^2 [10,13]^3 [11,13]^2 [11,14]^3 [11,15]^2 [12,15]^7 [12,16] [13,16]^2 [14,16]^6
I16p [1,2]^4 [1,3]^5 [2,4] [2,5]^2 [2,6]^2 [3,6]^2 [3,7]^2 [4,7]^5 [4,8]^3 [5,8]^2 [5,9]^5 [6,9]^3 [6,10]^2 [7,10]^2 [8,10]^4 [9,11] [10,12] [11,12]^6 [11,13]^2 [12,13]^2 [13,14]^4 [13,15] [14,15]^2 [14,16]^3 [15,16]^6
I16q [1,2]^8 [1,3] [2,4] [3,4]^2 [3,5]^4 [3,6]^2 [4,6]^3 [4,7]^3 [5,7]^5 [6,8]^2 [6,9]^2 [7,9] [8,10]^6 [8,11] [9,11]^6 [10,12]^3 [11,12]^2 [12,13]^3 [12,14] [13,14]^4 [13,15]^2 [14,15] [14,16]^3 [15,16]^6
I16r [1,2]^4 [1,3]^5 [2,3] [2,4] [2,5]^3 [3,5]^3 [4,6]^4 [4,7]^2 [4,8]^2 [5,8]^3 [6,8]^3 [6,9]^2 [7,9]^2 [7,10]^4 [7,11] [8,11] [9,11]^2 [9,12]^3 [10,12]^4 [10,13] [11,13]^3 [11,14]^2 [12,14]^2 [13,15]^5 [14,16]^5 [15,16]^4
I16s [1,2] [1,3]^8 [2,3] [2,4]^4 [2,5]^3 [4,5] [4,6] [4,7]^3 [5,7] [5,8]^2 [5,9] [5,10] [6,10]^7 [6,11] [7,11]^4 [7,12] [8,13]^5 [8,14]^2 [9,14]^7 [9,15] [10,15] [11,15]^4 [12,15]^3 [12,16]^5 [13,16]^4
I16t [1,2]^6 [1,3]^3 [2,3] [2,4]^2 [3,4]^3 [3,5]^2 [4,5]^2 [4,6]^2 [5,7]^5 [6,7]^4 [6,8] [6,9]^2 [8,9]^7 [8,10] [10,11]^6 [10,12]^2 [11,12] [11,13]^2 [12,13] [12,14]^5 [13,15]^5 [13,16] [14,16]^4 [15,16]^4
I16u [1,2]^6 [1,3] [1,4]^2 [2,4]^3 [3,4] [3,5]^7 [4,5] [4,6]^2 [5,6] [6,7]^3 [6,8] [6,9]^2 [7,9]^6 [8,9] [8,10]^3 [8,11] [8,12]^3 [10,12] [10,13] [10,14]^4 [11,14]^4 [11,15]^4 [12,15]^5 [13,16]^8 [14,16]
I18a [1,2]^3 [1,3] [1,4]^5 [2,5]^6 [3,5]^2 [3,6]^6 [4,6]^2 [4,7]^2 [5,7] [6,8] [7,8]^5 [7,9] [8,9]^3 [9,10]^3 [9,11]^2 [10,12]^6 [11,13]^5 [11,14]^2 [12,14]^3 [13,14] [13,15]^3 [14,16]^3 [15,16]^5 [15,17] [16,18] [17,18]^8
I18b [1,2]^3 [1,3]^6 [2,3]^3 [2,4]^3 [4,5]^2 [4,6] [4,7]^3 [5,7]^2 [5,8] [5,9]^4 [6,9] [6,10]^5 [6,11]^2 [7,11]^3 [7,12] [8,12]^5 [8,13]^3 [9,13]^3 [9,14] [10,15]^3 [10,16] [11,16]^4 [12,16]^3 [13,17]^3 [14,17]^6 [14,18]^2 [15,18]^6 [16,18]
I18c [1,2]^5 [1,3]^3 [1,4] [2,4]^2 [2,5]^2 [3,5]^3 [3,6]^3 [4,7]^6 [5,7]^3 [5,8] [6,8]^5 [6,9] [8,9]^3 [9,10]^4 [9,11] [10,11] [10,12]^4 [11,12]^5 [11,13]^2 [13,14]^5 [13,15] [13,16] [14,16]^4 [15,16]^3 [15,17]^3 [15,18]^2 [16,18] [17,18]^6
I18d [1,2]^8 [1,3] [2,4] [3,4]^5 [3,5]^3 [4,5] [4,6]^2 [5,6]^3 [5,7]^2 [6,7] [6,8]^3 [7,9]^6 [8,9] [8,10]^2 [8,11]^3 [9,11]^2 [10,11]^4 [10,12] [10,13]^2 [12,14]^5 [12,15]^3 [13,15]^6 [13,16] [14,16] [14,17]^3 [16,17]^2 [16,18]^5 [17,18]^4
I18e [1,2] [1,3]^6 [1,4]^2 [2,4]^5 [2,5]^2 [2,6] [3,6] [3,7]^2 [4,8] [4,9] [5,9]^7 [6,9] [6,10]^3 [6,11]^3 [7,11]^3 [7,12]^3 [7,13] [8,13]^8 [10,14]^6 [11,14]^2 [11,15] [12,15]^3 [12,16] [12,17]^2 [14,17] [15,17]^5 [16,18]^8 [17,18]
I18f [1,2]^5 [1,3]^4 [2,3]^3 [2,4] [3,4]^2 [4,5]^6 [5,6]^2 [5,7] [6,7] [6,8]^6 [7,8]^3 [7,9]^4 [9,10]^2 [9,11]^3 [10,11]^3 [10,12]^2 [10,13]^2 [11,13]^3 [12,13]^3 [12,14]^2 [12,15]^2 [13,15] [14,15]^2 [14,16]^4 [14,17] [15,17]^4 [16,18]^5 [17,18]^4
I18g [1,2] [1,3]^3 [1,4]^5 [2,4]^2 [2,5]^6 [3,5]^3 [3,6]^3 [4,6] [4,7] [6,7]^2 [6,8]^3 [7,8]^2 [7,9] [7,10] [7,11]^2 [8,11]^4 [9,11]^2 [9,12]^6 [10,12] [10,13]^2 [10,14]^5 [11,15] [12,15]^2 [13,15]^4 [13,16]^3 [14,16] [14,17]^3 [15,17]^2 [16,18]^5 [17,18]^4
I18h [1,2] [1,3]^8 [2,4]^6 [2,5]^2 [3,5] [4,5]^3 [5,6]^3 [6,7]^3 [6,8]^3 [7,8] [7,9]^2 [7,10]^2 [7,11] [8,11]^5 [9,11]^3 [9,12]^2 [9,13]^2 [10,13]^6 [10,14] [12,14]^5 [12,15]^2 [13,15] [14,15]^3 [15,16]^3 [16,17]^3 [16,18]^3 [17,18]^6
I18i [1,2]^3 [1,3]^5 [1,4] [2,4] [2,5]^5 [3,5] [3,6]^2 [3,7] [4,7]^3 [4,8] [4,9]^3 [5,9]^3 [6,9]^3 [6,10]^4 [7,10] [7,11]^4 [8,11]^2 [8,12]^6 [10,12]^2 [10,13]^2 [11,14] [11,15]^2 [12,15] [13,15] [13,16]^5 [13,17] [14,17]^8 [15,18]^5 [16,18]^4
I18j [1,2]^5 [1,3]^3 [1,4] [2,4]^4 [3,4]^3 [3,5]^3 [4,5] [5,6] [5,7]^2 [5,8] [5,9] [6,9]^6 [6,10]^2 [7,10]^2 [7,11]^5 [8,12]^7 [8,13] [9,13]^2 [10,13]^5 [11,13] [11,14]^3 [12,15]^2 [14,15] [14,16] [14,17]^4 [15,17]^5 [15,18] [16,18]^8
I18k [1,2]^8 [1,3] [2,4] [3,4]^4 [3,5]^4 [4,6]^4 [5,6]^2 [5,7]^3 [6,7]^3 [7,8] [7,9]^2 [8,9]^5 [8,10]^3 [9,10] [9,11] [10,11]^5 [11,12] [11,13]^2 [12,13]^4 [12,14]^4 [13,14]^2 [13,15] [14,15] [14,16]^2 [15,16]^5 [15,17]^2 [16,18]^2 [17,18]^7
I18l [1,2]^8 [1,3] [2,4] [3,4]^3 [3,5]^2 [3,6]^3 [4,6] [4,7]^4 [5,7]^5 [5,8] [5,9] [6,9]^5 [8,9]^2 [8,10]^2 [8,11]^4 [9,11] [10,11]^2 [10,12]^4 [10,13] [11,13]^2 [12,13] [12,14]^4 [13,14]^3 [13,15]^2 [14,15]^2 [15,16]^3 [15,17]^2 [16,17]^2 [16,18]^4 [17,18]^5
I18m [1,2]^5 [1,3]^4 [2,3]^3 [2,4] [3,5]^2 [4,5]^6 [4,6]^2 [5,6] [6,7]^2 [6,8] [6,9]^3 [7,9]^2 [7,10]^3 [7,11]^2 [8,11]^2 [8,12]^4 [8,13]^2 [9,13]^4 [10,13] [10,14]^5 [11,14]^4 [11,15] [12,15]^5 [13,16]^2 [15,17]^3 [16,17]^2 [16,18]^5 [17,18]^4
I18n [1,2] [1,3]^6 [1,4]^2 [2,4]^6 [2,5]^2 [3,6]^3 [4,7] [5,7]^3 [5,8]^3 [5,9] [6,9]^3 [6,10]^3 [7,10]^4 [7,11] [8,11]^3 [8,12]^3 [9,13]^2 [9,14]^3 [10,14] [10,15] [11,15]^5 [12,15] [12,16]^5 [13,17]^7 [14,17]^2 [14,18]^3 [15,18]^2 [16,18]^4
I18o [1,2]^3 [1,3]^6 [2,3]^2 [2,4]^4 [3,4] [4,5]^3 [4,6] [5,6]^2 [5,7]^4 [6,8]^2 [6,9] [6,10]^2 [6,11] [7,11]^5 [8,11]^2 [8,12] [8,13]^4 [9,13]^3 [9,14]^5 [10,15]^3 [10,16]^4 [11,16] [12,16]^3 [12,17]^5 [13,17]^2 [14,17]^2 [14,18]^2 [15,18]^6 [16,18]
I18p [1,2]^8 [1,3] [2,3] [3,4]^2 [3,5] [3,6] [3,7]^3 [4,7]^6 [4,8] [5,8]^2 [5,9] [5,10]^5 [6,11]^8 [8,11] [8,12]^5 [9,12] [9,13]^2 [9,14]^5 [10,14] [10,15]^3 [12,16]^3 [13,16]^2 [13,17]^5 [14,17]^3 [15,17] [15,18]^5 [16,18]^4
I18q [1,2]^5 [1,3]^2 [1,4]^2 [2,4]^4 [3,4]^3 [3,5] [3,6]^3 [5,6]^2 [5,7]^2 [5,8]^2 [5,9]^2 [6,9]^3 [6,10] [7,11]^7 [8,11] [8,12]^4 [8,13]^2 [9,13]^4 [10,13]^3 [10,14]^2 [10,15]^2 [10,16] [11,16] [12,16]^4 [12,17] [14,17]^7 [15,17] [15,18]^6 [16,18]^3
I18r [1,2]^6 [1,3] [1,4]^2 [2,4]^3 [3,4]^4 [3,5]^3 [3,6] [5,6]^4 [5,7]^2 [6,7]^3 [6,8] [7,8]^4 [8,9]^2 [8,10]^2 [9,10] [9,11]^6 [10,11]^3 [10,12] [10,13]^2 [12,13]^3 [12,14]^5 [13,15]^4 [14,16]^4 [15,17]^5 [16,18]^5 [17,18]^4
I18s [1,2]^4 [1,3]^2 [1,4]^3 [2,4]^5 [3,5]^7 [4,6] [5,7]^2 [6,7]^7 [6,8] [8,9]^3 [8,10]^5 [9,10]^4 [9,11] [9,12] [11,12]^2 [11,13]^2 [11,14]^4 [12,14]^5 [12,15] [13,16]^7 [15,16]^2 [15,17]^3 [15,18]^3 [17,18]^6
I18t [1,2]^8 [1,3] [2,3] [3,4]^2 [3,5]^4 [3,6] [4,6]^6 [4,7] [5,7]^5 [6,8]^2 [7,8] [7,9]^2 [8,9]^3 [8,10]^3 [9,10]^4 [10,11]^2 [11,12]^2 [11,13]^5 [12,13] [12,14]^6 [13,15]^3 [14,15]^3 [15,16]^3 [16,17]^3 [16,18]^3 [17,18]^6
I18u [1,2]^6 [1,3]^3 [2,3]^3 [3,4]^3 [4,5]^4 [4,6]^2 [5,6]^2 [5,7]^3 [6,7] [6,8]^4 [7,9]^4 [7,10] [8,11]^5 [9,12] [9,13]^4 [10,13]^5 [10,14]^3 [11,14]^4 [12,14] [12,15] [12,16]^6 [14,16] [15,16] [15,17]^4 [15,18]^3 [16,18] [17,18]^5
I18v [1,2]^6 [1,3] [1,4]^2 [2,4] [2,5]^2 [3,5]^6 [3,6]^2 [4,7]^5 [4,8] [5,8] [6,8] [6,9]^2 [6,10]^4 [7,10]^3 [7,11] [8,11]^4 [8,12]^2 [9,12]^4 [9,13]^3 [10,14] [10,15] [11,15]^4 [12,15] [12,16]^2 [13,16]^3 [13,17]^3 [14,17]^6 [14,18]^2 [15,18]^3 [16,18]^4
I18w [1,2] [1,3]^6 [1,4]^2 [2,4]^6 [2,5]^2 [3,5]^3 [4,6] [5,6]^2 [5,7]^2 [6,7] [6,8] [6,9]^4 [7,9]^2 [7,10]^4 [8,10] [8,11]^5 [8,12]^2 [9,12]^3 [10,12]^4 [11,13]^2 [11,14]^2 [13,14]^2 [13,15]^3 [13,16]^2 [14,17]^5 [15,17]^4 [15,18]^2 [16,18]^7
I18x [1,2]^4 [1,3]^5 [2,3] [2,4]^4 [3,5]^3 [4,5]^2 [4,6]^3 [5,6]^2 [5,7] [5,8] [6,8]^4 [7,8]^3 [7,9]^3 [7,10]^2 [8,10] [9,10]^3 [9,11]^3 [10,11] [10,12]^2 [11,12] [11,13]^2 [11,14]^2 [12,15]^6 [13,15]^2 [13,16]^5 [14,16]^2 [14,17]^4 [14,18] [15,18] [16,18]^2 [17,18]^5
I18y [1,2]^3 [1,3]^6 [2,3]^3 [2,4]^2 [2,5] [4,6]^4 [4,7]^3 [5,7]^2 [5,8]^4 [5,9]^2 [6,9]^2 [6,10]^3 [7,10]^3 [7,11] [8,11]^5 [9,11] [9,12]^2 [9,13]^2 [10,13]^3 [11,13]^2 [12,13] [12,14] [12,15]^5 [13,15] [14,16]^8 [15,16] [15,17] [15,18] [17,18]^8
I20a [1,2]^2 [1,3]^2 [1,4]^4 [1,5] [2,5]^3 [2,6]^4 [3,6]^4 [3,7]^3 [4,7]^5 [5,8] [5,9]^4 [6,10] [7,10] [8,10]^3 [8,11]^5 [9,12]^5 [10,13]^4 [11,13]^2 [11,14] [11,15] [12,15]^3 [12,16] [13,16]^2 [13,17] [14,17]^6 [14,18]^2 [15,18] [15,19]^4 [16,19]^5 [16,20] [17,20]^2 [18,20]^6
I20b [1,2]^3 [1,3] [1,4]^5 [2,4]^2 [2,5] [2,6]^3 [3,6]^6 [3,7]^2 [4,7]^2 [5,7]^4 [5,8]^4 [7,8] [8,9]^2 [8,10]^2 [9,11]^7 [10,11] [10,12] [10,13]^5 [11,13] [12,13]^3 [12,14]^5 [14,15] [14,16]^3 [15,16]^2 [15,17]^6 [16,17]^2 [16,18] [16,19] [17,19] [18,19]^3 [18,20]^5 [19,20]^4
I22 [1,2]^2 [1,3]^6 [1,4] [2,4]^5 [2,5]^2 [3,5]^3 [4,5]^2 [4,6] [5,6]^2 [6,7]^4 [6,8]^2 [7,8] [7,9] [7,10]^2 [7,11] [8,11]^6 [9,11]^2 [9,12]^6 [10,13]^7 [12,14]^3 [13,14] [13,15] [14,15] [14,16]^4 [15,17]^7 [16,18]^3 [16,19]^2 [17,19]^2 [18,20]^6 [19,20]^3 [19,21] [19,22] [21,22]^8

## The binary decimic

Complete list of degrees of basic invariants: degree 2 (once), 4 (once), 6 (4 times), 8 (5 times), 9 (5 times), 10 (8 times), 11 (8 times), 12 (12 times), 13 (15 times), 14 (13 times), 15 (19 times), 16 (5 times), 17 (5 times), 18 (once), 19 (2 times), 21 (2 times) for a total of 106 basic invariants.

(This list agrees with Sylvester for degrees less than 17. Sylvester predicted 3 basic invariants of degree 17 and none of degree higher than 17 for a total of 99 basic invariants. Olver (p. 40) reports 104 invariants. The existence of basic invariants of degree 21 seems to be new. That the list is complete follows from the result by aeb & M. Popoviciu that there is a hsop with degrees 2, 4, 6, 6, 8, 9, 10, 14.)

Numbers of basic invariants and covariants (the latter possibly incomplete):

d\o 0 2 4 6 8 10 12 14 16 18 20 22 24 26 # cum
1 - - - - - 1 - - - - - - - - 1 1
2 1 - 1 - 1 - 1 - 1 - - - - - 5 6
3 - 1 - 2 1 1 2 1 1 1 1 - 1 - 12 18
4 1 - 3 1 3 3 2 3 1 2 1 1 - 1 22 40
5 - 3 3 4 5 4 5 2 4 - 2 - - - 32 72
6 4 2 5 8 6 8 2 4 - 1 - - - - 40 112
7 - 7 10 8 12 2 4 - 1 - - - - - 44 156
8 5 8 11 15 4 7 - 1 - - - - - - 51 207
9 5 13 19 8 7 - 1 - - - - - - - 53 260
10 8 20 13 13 - 1 - - - - - - - - 55 315
11 8 18 21 - 1 - - - - - - - - - 48 363
12 12 30 1 2 - - - - - - - - - - 45 408
13 15 16 2 - - - - - - - - - - - 33 441
14 13 17 - - - - - - - - - - - - 30 471
15 19 - 1 - - - - - - - - - - - 20 491
16 5 3 - - - - - - - - - - - - 8 499
17 5 - - - - - - - - - - - - - 5 504
18 1 1 - - - - - - - - - - - - 2 506
19 2 - - - - - - - - - - - - - 2 508
20 - - - - - - - - - - - - - - - 508
21 2 - - - - - - - - - - - - - 2 510

(Partial results computed by Tom Hagedorn, aeb.)

## The binary duodecimic

A, possibly incomplete, list of basic invariants: degree 2 (once), 3 (once), 4 (2 times), 5 (2 times), 6 (4 times), 7 (5 times), 8 (7 times), 9 (9 times), 10 (14 times), 11 (15 times), 12 (19 times), 13 (18 times), 14 (12 times), 15 (2 times), 16 (once), 17 (once), for a total of 113 basic invariants known. Any further basic invariant, if there is one, has degree at least 30.

(This list agrees with Sylvester for degrees less than 15. Sylvester predicted none of degree higher than 14 for a total of 109 basic invariants.)

Numbers of basic invariants and covariants (possibly incomplete):

d\o 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 # cum
1 - - - - - - 1 - - - - - - - - - - - 1 1
2 1 - 1 - 1 - 1 - 1 - 1 - - - - - - - 6 7
3 1 - 1 1 2 1 2 2 1 2 1 1 1 1 - 1 - - 18 25
4 2 - 3 2 4 3 4 4 3 4 2 3 1 2 1 1 - 1 40 65
5 2 2 5 6 7 8 6 9 5 6 3 5 1 2 - 1 - - 68 133
6 4 4 9 11 12 14 10 12 3 6 1 1 - - - - - - 87 220
7 5 10 15 20 18 21 9 9 1 1 - - - - - - - - 109 329
8 7 16 24 29 21 21 1 1 1 - - - - - - - - - 121 450
9 9 28 33 37 15 1 1 1 - - - - - - - - - - 125 575
10 14 39 41 30 1 1 1 - - - - - - - - - - - 127 702
11 15 53 40 2 1 1 - - - - - - - - - - - - 112 814
12 19 56 8 1 1 - - - - - - - - - - - - - 85 899
13 18 44 2 1 - - - - - - - - - - - - - - 65 964
14 12 5 1 - - - - - - - - - - - - - - - 18 982
15 2 2 - - - - - - - - - - - - - - - - 4 986
16 1 1 - - - - - - - - - - - - - - - - 2 988
17 1 - - - - - - - - - - - - - - - - - 1 989

## Order bound

Grace & Young §271 say that the maximum order of an irreducible covariant for n = 2a+m with m < n/2 is at most (and probably, for n not 3, is) (a–1)n+2m+2. This is slightly better than a similar bound by Jordan.

## Terminology

The terms used are not very standard. For n=7 one sees septimic, septic, seventhic and more. For n=8 one also finds octic, for n=10 decic or ten-ic, for n=12 twelvic. Maybe originally these terms were derived from the Latin ordinalia. So: quartic, quintic, sextic, septimic, octavic, nonic, decimic.

## References

Leonid Bedratyuk, On complete system of invariants for the binary form of degree 7, J. Symb. Comput. 42 (2007) 935-947.

Leonid Bedratyuk, A complete minimal system of covariants for the binary form of degree 7, J. Symb. Comput. 44 (2009) 211-220.

Leonid Bedratyuk, On complete system of covariants for the binary form of degree 8, arXiv 0612113v1 (2006).

A. E. Brouwer & M. Popoviciu, The invariants of the binary nonic, J. Symb. Comput. 45 (2010) 709-720.

A. E. Brouwer & M. Popoviciu, The invariants of the binary decimic, J. Symb. Comput. 45 (2010) 837-843.

Holger Cröni, Zur Berechnung von Kovarianten von Quantiken, Dissertation, Univ. des Saarlandes, Saarbrücken, 2002.

Jacques Dixmier, Quelques résultats et conjectures concernant les séries de Poincaré des invariants de formes binaires, pp. 127-160 in: Séminaire d'algèbre Paul Dubreil et Marie-Paule Malliavin (1983-1984), Springer LNM 1146, 1985.

J. Dixmier & D. Lazard, Le nombre minimum d'invariants fondamentaux pour les formes binaires de degré 7, Portug. Math. 43 (1986) 377-392. Also J. Symb. Comput. 6 (1988) 113-115.

F. von Gall, Das vollständige Formensystem der binären Form 7ter Ordnung, Math. Ann. 31 (1888) 318-336.

F. von Gall, Das vollständige Formensystem einer binären Form achter Ordnung, Math. Ann. 17 (1880) 31-51, 139-152, 456.

P. Gordan, Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist, Journ. f. Math. 69 (1868) 323-354.

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

J. Hammond, A simple proof of the existence of irreducible invariants of degrees 20 and 30 for the binary seventhic, Math. Ann. 36 (1890) 255-261.

Joseph P. S. Kung & Gian-Carlo Rota, The Invariant Theory of Binary Forms, Bull. Amer. Math. Soc. 10 (1984) 27-85.

P. Olver, Classical Invariant Theory, LMS Student Texts 44, Cambridge, 1999.

Issai Schur, Vorlesungen über Invariantentheorie, Springer, 1968.

T. Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89 (1967) 1022-1046.

J. J. Sylvester & F. Franklin, Tables of generating functions and groundforms for the binary quantics of the first ten orders, Amer. J. Math. 2 (1879) 223-251.

J. J. Sylvester & F. Franklin, Tables of the generating functions and groundforms for the binary duodecimic, with some general remarks, and tables of the irreducible syzygies of certain quantics, Amer. J. Math. 4 (1881) 41-61.