In mathematics, a '''ternary quartic form''' is a degree 4 homogeneous polynomial in three variables.
==Hilbert's theorem==
{{harvs|txt|last=Hilbert|year=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==
thumb|250px|right|Table 2 from Noether's dissertation {{harv|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) {{harv|Dixmier|1987}}, together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by {{harvtxt|Shioda|1967}}. {{harvtxt|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. {{harv|Dolgachev|2012|loc=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.
==See also==
*Ternary cubic *Invariants of a binary form
==References==
*{{Citation | last1=Cohen | first1=Teresa | title=Investigations on the Plane Quartic | year=1919 | journal=American Journal of Mathematics | issn=0002-9327 | volume=41 | issue=3 | pages=191–211 | jstor=2370332 | doi=10.2307/2370332| hdl=2027/mdp.39015079994953 | hdl-access=free }} *{{Citation | last1=Dixmier | first1=Jacques | authorlink1=Jacques Dixmier | title=On the projective invariants of quartic plane curves | doi=10.1016/0001-8708(87)90010-7 | doi-access=free |mr=888630 | year=1987 | journal=Advances in Mathematics | issn=0001-8708 | volume=64 | issue=3 | pages=279–304}} *{{Citation | last1=Dolgachev |first1=Igor |title=Classical Algebraic Geometry : A Modern View |year=2012 |isbn=978-1-1070-1765-8 |publisher=Cambridge University Press |url=https://books.google.com/books?id=g9GLXmR9qg0C&pg=266 }} *{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Darstellung definiter Formen als Summe von Formenquadraten | year=1888 | journal=Mathematische Annalen | issn=0025-5831 | volume=32 | issue=3 | pages=342–350 | doi=10.1007/BF01443605| url=https://zenodo.org/record/1428214 }} *{{citation|last=Noether|first=Emmy|title=Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms)|journal=Journal für die reine und angewandte Mathematik|volume=134|year=1908|pages=23–90 and two tables|url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261200|url-status=dead|archiveurl=https://web.archive.org/web/20130308102907/http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261200|archivedate=2013-03-08}}. *{{Citation | last1=Salmon | first1=George | title=A treatise on the higher plane curves | orig-date=1852 | url=https://archive.org/details/atreatiseonhigh01caylgoog | publisher=Hodges, Foster and Figgis | isbn=978-1-4181-8252-6 |mr=0115124 | year=1879}} *{{Citation | last1=Shioda | first1=Tetsuji | title=On the graded ring of invariants of binary octavics |mr=0220738 | year=1967 | journal=American Journal of Mathematics | issn=0002-9327 | volume=89 | issue=4 | pages=1022–1046 | doi=10.2307/2373415| jstor=2373415 }} *{{Citation | last1=Thomsen | first1=H. Ivah | title=Some Invariants of the Ternary Quartic | year=1916 | journal=American Journal of Mathematics | issn=0002-9327 | volume=38 | issue=3 | pages=249–258 | jstor=2370450 | doi=10.2307/2370450}}
==External links==
*[http://www.win.tue.nl/~aeb/math/ternary_quartic.html Invariants of the ternary quartic]
Category:Invariant theory