Cubic form

A homogeneous polynomial of degree 3 in several variables with coefficients in some fixed field or ring. Let $K$ be a field and $F_3(x_0,\ldots,x_n)$ a cubic form with coefficients in $K$ (one calls it a cubic form over $K$). The equation $$ F_3(x_0,\ldots,x_n) = 0 $$ defines a cubic hypersurface in the projective space $\mathbf{P}^n(K)$, so that the algebraic-geometric theory of cubic forms over an algebraically closed field $K$ is reduced to the theory of cubic hypersurfaces (see [1]).

The arithmetic theory of cubic forms over number fields (and their rings of integers) is still (1987) rather poorly developed in comparison with the rich and meaningful arithmetic theory of quadratic forms. For cubic forms in two variables, the arithmetic theory is just the theory of cubic extensions of number fields (see [2]). For cubic forms in three variables, it is part of the arithmetic theory of elliptic curves (see [3]). In particular, examples are known of cubic forms in three variables that violate the Hasse principle. The same holds for cubic forms in four variables (see [1], [4], [6]). There is no general theory at all for cubic forms in a larger number of variables.

The purely algebraic theory of cubic forms contains, in addition to results concerning the structure of sets of points on cubic hypersurfaces, various results pertaining to the classical theory of invariants. Indeed, the structure of the algebra of (absolute) invariants of a cubic form in two or three variables is known; in these cases the algebra has no syzygies — it is the algebra of polynomials in one (degree 4) and two (degrees 4 and 6) algebraically independent homogeneous generators. If the number of variables exceeds 4, the algebra in question contains syzygies [5] and its structure is very complicated. H. Poincaré investigated the orbits, stabilizers and canonical representatives, and also the families of orbits under the natural action of the group $\mathrm{SL}_n$ on the space of all cubic forms in three and four variables [6].

References

[1]	Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1986) (Translated from Russian) MR0833513 Zbl 0582.14010
[2]	B.N. Delone, D.K. Faddeev, "Theory of irrationalities of the third degree" Trudy Mat. Inst. Akad. Nauk SSSR , 11 (1940) (In Russian) MR0160744 Zbl 0061.09001
[3]	J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291 MR0199150 Zbl 0138.27002
[4]	B. Segree, "On the rational solutions of homogeneous cubic equations in four variables" Math. Notae (Univ. Rosario) , 11 (1951) pp. 1–68
[5]	V.G. Kats, V.L. Popov, E.B. Vinberg, "Sur les groupes linéaires algébriques dont l'algèbre des invariants est libre" Acad. Sci. Paris , 283 (1976) pp. 875–878 (English summary)
[6]	H. Poincaré, , Ouevres , 2 , Moscow (1972) pp. 819–900 (In Russian) Zbl 54.0036.04