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Algebraic form

In the mathematics of the nineteenth century, an important role was played by the algebraic forms that generalise quadratic forms to degrees 3 and more, also known as quantics. These are the homogeneous polynomials of a fixed degree n in a fixed number m of variables: the m-ary n-ics as it would then have been put (with complex number coefficients understood).

The two obvious areas where these would be applied were projective geometry, and number theory (less then in fashion). The geometric use was connected with invariant theory. There is a general linear group acting on any given space of quantics, and this group action is potentially a fruitful way to classify algebraic varieties (for example cubic hypersurfaces in a given number of variables).

In more modern language the spaces of quantics are identified with the symmetric tensors of a given degree constructed from the tensor powers of a vector space V of dimension m. (This is straightforward provided we work over a field of characteristic zero). That is, we take the n-fold tensor product of V with itself and take the subspace invariant under the symmetric group as it permutes factors. This definition specifies how GL(V) will act.

It would be a possible direct method in algebraic geometry, to study the orbitss of this action. More precisely the orbits for the action on the projective space formed from the vector space of symmetric tensors. The construction of invariants would be the theory of the co-ordinate ring of the 'space' of orbits, assuming that 'space' exists. No direct answer to that was given, until the geometric invariant theory of David Mumford; so the invariants of quantics were studied directly. Heroic calculations were performed, in an era leading up to the work of David Hilbert on the qualitative theory.

For algebraic forms with integer coefficients, generalisations of the classical results on quadratic forms to forms of higher degree motivated much investigation.