Quaternary cubic

In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space.

Invariants

Salmon (1860) and Clebsch (1861, 1861b) 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 Edge (1980) 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 in 1851, and proven 10 years later by Clebsch. The union of the 5 planes where these 5 linear forms vanish is called the Sylvester pentahedron.

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See also

References
    • Clebsch, A. (1861), "Zur Theorie der algebraischer Flächen", Journal für die reine und angewandte Mathematik, 58: 93–108, ISSN 0075-4102
    • Clebsch, A. (1861), "Ueber eine Transformation der homogenen Funktionen dritter Ordnung mit vier Veränderlichen", Journal für die reine und angewandte Mathematik, 58: 109–126, doi:10.1515/crll.1861.58.109, ISSN 0075-4102
    • Edge, W. L. (1980), "The Discriminant of a Cubic Surface", Proceedings of the Royal Irish Academy, Royal Irish Academy, 80A (1): 75–78, ISSN 0035-8975, JSTOR 20489083
    • Salmon, George (1860), "On Quaternary Cubics", Philosophical Transactions of the Royal Society, The Royal Society, 150: 229–239, doi:10.1098/rstl.1860.0015, ISSN 0080-4614, JSTOR 108770
    • Schmitt, Alexander (1997), "Quaternary cubic forms and projective algebraic threefolds", L'Enseignement Mathématique. Revue Internationale. IIe Série, 43 (3): 253–270, ISSN 0013-8584, MR 1489885
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