Ternary quartic

{{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.

File:Emmy Noether - Table of invariants 2.jpg

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}}

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.

• Ternary cubic
• Invariants of a binary form

• {{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 | origyear=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}}

• [http://www.win.tue.nl/~aeb/math/ternary_quartic.html Invariants of the ternary quartic]

Category:Invariant theory