Invariants of the ternary quartic

The ternary quartic form (three variables, homogeneous of degree 4) with complex coefficients has under SL(3,C) seven algebraically independent invariants, of degrees 3, 6, 9, 12, 15, 18, 27, as was proved by Dixmier (1987).

Shioda (1967) computed the Poincaré series P(t)/Q(t) with numerator

P(t) = 1 + t9 + t12 + t15 + 2t18 + 3t21 + 2t24 + 3t27 + 4t30 + 3t33 + 4t36 + 4t39 + ... + t75

(where ti and t75–i have equal coefficients) and denominator

Q(t) = (1–t3)(1–t6)(1–t9)(1–t12)(1–t15)(1–t18)(1–t27).

From this he conjectured that there should be six more basic invariants, of degrees 9, 12, 15, 18, 21, 21, and this is indeed the case.

The invariants of degree at most 18 can be found in Salmon (1879). (Those of degrees 15 and 18 had been computed by J.J. Walker.) So, it remains to give the invariants of degree 21.

The invariant of degree 27 is the determinant. The remaining invariants can be given as follows in bracket notation.

degree bracket monomial
 
3 [1,2,3][1,2,3][1,2,3][1,2,3]
6 [1,2,3][1,2,4][1,2,5][1,3,5][2,4,6][3,4,6][3,5,6][4,5,6]
9 [1,2,3][1,2,4][1,2,5][1,3,5][2,3,6][3,4,6][4,6,7][4,7,8][5,7,9][5,8,9][6,8,9][7,8,9]
9 [1,2,3][1,2,3][1,4,5][1,4,5][2,5,6][2,5,6][3,7,8][3,7,8][4,7,9][4,7,9][6,8,9][6,8,9]
12 [1,2,3][1,2,3][1,4,5][1,4,5][2,6,7][2,6,7][3,8,9][3,8,9][4,6,10][4,6,10][5,8,11][5,8,11]
[7,9,12][7,9,12][10,11,12][10,11,12]
12 [1,2,3][1,2,3][1,4,5][1,4,5][2,6,7][2,6,7][3,8,9][3,8,9][4,6,10][4,6,10][5,8,11][5,8,11]
[7,9,12][7,10,12][9,11,12][10,11,12]
15 [1,2,3][1,2,3][1,4,5][1,4,5][2,4,6][2,4,6][3,5,7][3,5,7][6,8,9][6,8,9][7,10,11][7,10,11]
[8,12,13][8,12,13][9,14,15][9,14,15][10,12,14][10,12,14][11,13,15][11,13,15]
15 [1,2,3][1,2,3][1,4,5][1,4,5][2,4,6][2,4,6][3,5,7][3,5,7][6,8,9][6,8,9][7,10,11][7,11,12]
[8,11,13][8,12,13][9,14,15][9,14,15][10,12,14][10,12,14][10,13,15][11,13,15]
18 [1,2,3][1,2,3][1,4,5][1,4,5][2,4,6][2,4,6][3,5,7][3,5,7][6,8,9][6,8,9][7,10,11][7,10,11]
[8,12,13][8,12,13][9,14,15][9,14,15][10,16,17][10,16,17][11,16,18][11,16,18]
[12,17,18][12,17,18][13,14,15][13,14,15]
18 [1,2,3][1,2,3][1,4,5][1,4,5][2,4,6][2,4,6][3,5,7][3,5,7][6,8,9][6,8,9][7,10,11][7,10,11]
[8,12,13][8,12,13][9,12,15][9,14,15][10,16,17][10,16,17][11,16,18][11,16,18]
[12,17,18][14,17,18][13,14,15][13,14,15]
21 [1,2,3][1,2,3][1,4,5][1,4,5][2,4,6][2,4,6][3,5,7][3,5,7][6,8,9][6,8,9][7,10,11][7,10,11]
[8,12,13][8,12,13][9,14,15][9,14,15][10,16,17][10,16,17][11,16,18][11,16,18]
[12,17,18][12,17,18][13,19,20][13,19,20][14,19,21][14,19,21][15,20,21][15,20,21]
21 [1,2,3][1,2,3][1,4,5][1,4,5][2,4,6][2,4,6][3,5,7][3,5,7][6,8,9][6,8,9][7,10,11][7,10,11]
[8,12,13][8,12,13][9,14,15][9,14,15][10,16,17][10,16,17][11,16,18][11,12,18]
[12,17,18][16,17,18][13,19,20][13,19,20][14,19,21][14,19,21][15,20,21][15,20,21]

aeb, 2001-04-15