Ternary quartic

In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.

Hilbert's theorem

Hilbert (1888) showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.

Invariant theory

Table 2 from Noether's dissertation (Noether 1908) on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables x and u. The horizontal direction of the table lists the invariants with increasing grades in x, while the vertical direction lists them with increasing grades in u.

The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) (Dixmier 1987), together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by Shioda (1967). Salmon (1879) discussed the invariants of order up to about 15.

The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an inflection bitangent. (Dolgachev 2012, 6.4)

Catalecticant

The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms.

References

Cohen, Teresa (1919), "Investigations on the Plane Quartic", American Journal of Mathematics (The Johns Hopkins University Press) 41 (3): 191–211, ISSN 0002-9327, JSTOR 2370332
Dixmier, Jacques (1987), "On the projective invariants of quartic plane curves", Advances in Mathematics 64 (3): 279–304, doi:10.1016/0001-8708(87)90010-7, ISSN 0001-8708, MR 888630
Hilbert, David (1888), "Ueber die Darstellung definiter Formen als Summe von Formenquadraten", Mathematische Annalen (Springer Berlin / Heidelberg) 32: 342–350, doi:10.1007/BF01443605, ISSN 0025-5831
Noether, Emmy (1908), "Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms)", Journal für die reine und angewandte Mathematik 134: 23–90 and two tables .
Salmon, George (1879) [1852], A treatise on the higher plane curves, Hodges, Foster and Figgis, ISBN 978-1-4181-8252-6, MR 0115124
Shioda, Tetsuji (1967), "On the graded ring of invariants of binary octavics", American Journal of Mathematics 89: 1022–1046, doi:10.2307/2373415, ISSN 0002-9327, MR 0220738
Thomsen, H. Ivah (1916), "Some Invariants of the Ternary Quartic", American Journal of Mathematics (The Johns Hopkins University Press) 38 (3): 249–258, ISSN 0002-9327, JSTOR 2370450

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