In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a
cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...
in 3-dimensional projective space.
Invariants
and studied the ring of invariants of a quaternary cubic, which is a ring generated by invariants of
degrees 8, 16, 24, 32, 40, 100. The generators of degrees 8, 16, 24, 32, 40 generate a polynomial ring. The generator of degree 100 is a skew invariant, whose square is a polynomial in the other generators given explicitly by Salmon. Salmon also gave an explicit formula for the discriminant as a polynomial in the generators, though pointed out that the formula has a widely copied misprint in it.
Sylvester pentahedron
A generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cube roots of unity. This was conjectured by
Sylvester
Sylvester or Silvester is a name derived from the Latin adjective ''silvestris'' meaning "wooded" or "wild", which derives from the noun ''silva'' meaning "woodland". Classical Latin spells this with ''i''. In Classical Latin, ''y'' represented a ...
in 1851, and proven 10 years later by
Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Humboldt ...
. The union of the 5 planes where these 5 linear forms vanish is called the Sylvester pentahedron.
See also
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Ternary cubic In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
Invariant theory
The ternary cubic is one of the few cases of a form of degree greater than 2 in more than 2 variables whose ring of invariants was calcu ...
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Ternary quartic In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial 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 of quadratic form ...
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Invariants of a binary form In mathematical invariant theory, an invariant of a binary form is a polynomial in the coefficients of a binary form in two variables ''x'' and ''y'' that remains invariant under the special linear group acting on the variables ''x'' and ''y''.
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References
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*{{Citation , last1=Schmitt , first1=Alexander , title=Quaternary cubic forms and projective algebraic threefolds , url=http://retro.seals.ch/digbib/view?rid=ensmat-001:1997:43::125&id=hitlist , mr=1489885 , year=1997 , journal=L'Enseignement Mathématique , series=2e Série , issn=0013-8584 , volume=43 , issue=3 , pages=253–270
Invariant theory
Algebraic surfaces