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 theoryedit
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)
Catalecticantedit
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.
Dolgachev, Igor (2012), Classical Algebraic Geometry : A Modern View, Cambridge University Press, ISBN978-1-1070-1765-8
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
Noether, Emmy (1908), , Journal für die reine und angewandte Mathematik, 134: 23–90 and two tables, archived from the original on 2013-03-08.
Salmon, George (1879) [1852], A treatise on the higher plane curves, Hodges, Foster and Figgis, ISBN978-1-4181-8252-6, MR 0115124
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, JSTOR 2373415, MR 0220738
ternary, quartic, mathematics, ternary, quartic, form, degree, homogeneous, polynomial, three, variables, contents, hilbert, theorem, invariant, theory, catalecticant, also, references, external, linkshilbert, theorem, edithilbert, 1888, showed, that, positive. In mathematics a ternary quartic form is a degree 4 homogeneous polynomial in three variables Contents 1 Hilbert s theorem 2 Invariant theory 3 Catalecticant 4 See also 5 References 6 External linksHilbert s theorem editHilbert 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 edit nbsp 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 editThe 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 editTernary cubic Invariants of a binary formReferences editCohen Teresa 1919 Investigations on the Plane Quartic American Journal of Mathematics 41 3 191 211 doi 10 2307 2370332 hdl 2027 mdp 39015079994953 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 0888630 Dolgachev Igor 2012 Classical Algebraic Geometry A Modern View Cambridge University Press ISBN 978 1 1070 1765 8 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 Noether Emmy 1908 Uber die Bildung des Formensystems der ternaren biquadratischen Form On Complete Systems of Invariants for Ternary Biquadratic Forms Journal fur die reine und angewandte Mathematik 134 23 90 and two tables archived from the original on 2013 03 08 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 4 1022 1046 doi 10 2307 2373415 ISSN 0002 9327 JSTOR 2373415 MR 0220738 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 JSTOR 2370450External links editInvariants of the ternary quartic Retrieved from https en wikipedia org w index php title Ternary quartic amp oldid 1117832247, wikipedia, wiki, book, books, library,
