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In mathematics, a ternary quartic form is a degree 4
The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by . 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.
Invariants of the ternary quartic
Invariant theory
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...
in three variables.
Hilbert's theorem
showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares ofquadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s.
Invariant theory
The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by . 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.
Catalecticant
The catalecticant of a ternary quartic is theresultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...
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 formReferences
* * * * *. * * *{{Citation , last1=Thomsen , first1=H. Ivah , title=Some Invariants of the Ternary Quartic , year=1916 , journal=American Journal of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press.
History
The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...
, issn=0002-9327 , volume=38 , issue=3 , pages=249–258 , jstor=2370450 , doi=10.2307/2370450
External links
Invariants of the ternary quartic
Invariant theory