# 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

## History and further work

The basic work on invariants of the ternary quartic was done by Salmon (1879), using work by Cayley. He gives invariants of degree 3 (A, p. 264), 6 (B, p. 265), 9 (C1, C2, pp. 271--272), 12 (D1, D2, p. 272), 15 (E1, E2, pp. 273--274) and 18 (F1, F2, p. 274), where those of degree 15 and 18 had been computed by J. J. Walker. The invariants of order 21 were first given here.

Shioda (1967) determined the Poincaré series of the graded ring of invariants of ternary quartics, given above. He gave a detailed conjecture for the system of invariants.

Dixmier (1987) gave a homogeneous system of parameters, with degrees 3, 6, 9, 12, 15, 18, 27.

Brouwer (2001) found the invariants of degree 21. Ottaviani & Sernesi used the above expressions for the invariants to give an expression for the Lüroth invariant.

Ohno (2006?) found all invariants and syzygies, verifying Shioda's conjecture. David Kohel provided me with a copy of the unpublished preprint from 2007.

Kohel implemented the invariants given by Dixmier and Ohno in Magma. (The site is not accessible today.)

Armand Brumer (personal communications to David Kohel and me) programmed all the invariants in Maple at the end of 1996, and verified that his results agree with those of Ohno.

Elsenhans (2015) recomputed all invariants, again verifying Shioda's conjecture.

## References

J. Dixmier, On the projective invariants of quartic plane curves, Advances in Math. 64 (1987) 279-304.

Andreas-Stephan Elsenhans, Explicit computations of invariants of plane quartic curves, Journal of Symbolic Computation 68 (2015) 109-115. PDF

E. Noether, Über die Bildungen des Formensystems der ternären biquadratischen Form, J. Reine Angew. Math. 134 (1908) 23-90.

Toshiaki Ohno, The graded ring of invariants of ternary quartics I - generators and relations, preprint, 2007.

Giorgio Ottaviani & Edoardo Sernesi, On singular Lüroth quartics, Science China Mathematics 54 (August 2011) 1757-1766. (Preprint arXiv:0911.2101v2.)

G. Salmon, Higher plane curves, 3rd ed., 1879; reprinted by Chelsea, New York.

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

aeb, 2015-03-02