Cubic forms. algebra, geometry, arithmetic
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Fler böcker av Yu I Manin
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Old Password. A homogeneous polynomial of degree 3 in several variables with coefficients in some fixed field or ring.
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The arithmetic theory of cubic forms over number fields and their rings of integers is still 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 . For cubic forms in three variables, it is part of the arithmetic theory of elliptic curves see . In particular, examples are known of cubic forms in three variables that violate the Hasse principle.
Cubic forms : algebra, geometry, arithmetic
The same holds for cubic forms in four variables see  ,  , . 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  and its structure is very complicated.
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