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)

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.

See also

• Ternary cubic
• Invariants of a binary form

References

• Cohen, Teresa (1919), "Investigations on the Plane Quartic", American Journal of Mathematics 41 (3): 191–211, doi:10.2307/2370332, ISSN 0002-9327 
• 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 
• Dolgachev, Igor (2012), Classical Algebraic Geometry : A Modern View, Cambridge University Press, ISBN 978-1-1070-1765-8, https://books.google.com/books?id=g9GLXmR9qg0C&pg=266 
• Hilbert, David (1888), "Ueber die Darstellung definiter Formen als Summe von Formenquadraten", Mathematische Annalen 32 (3): 342–350, doi:10.1007/BF01443605, ISSN 0025-5831, https://zenodo.org/record/1428214 
• 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, archived from the original on 2013-03-08, https://web.archive.org/web/20130308102907/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261200 .
• Salmon, George (1879) [1852], A treatise on the higher plane curves, Hodges, Foster and Figgis, ISBN 978-1-4181-8252-6, https://archive.org/details/atreatiseonhigh01caylgoog 
• Shioda, Tetsuji (1967), "On the graded ring of invariants of binary octavics", American Journal of Mathematics 89 (4): 1022–1046, doi:10.2307/2373415, ISSN 0002-9327 
• Thomsen, H. Ivah (1916), "Some Invariants of the Ternary Quartic", American Journal of Mathematics 38 (3): 249–258, doi:10.2307/2370450, ISSN 0002-9327 

External links

• Invariants of the ternary quartic