TITLE: There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?
QUESTION [55 upvotes]: Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? If not, is it because:

It is not as interesting as quadratic forms?
It is so hard that no-one has yet written about such a theory?
It is already deeply infiltrated in math and only some smart people know about it?

REPLY [2 votes]: Here is an additional comment. Every homogeneous polynomial can be regarded as a symmetric tensor. Quadrics correspond to symmetric matrices; cubics correspond to order $3$ tensors; in general, a homogeneous polynomial of degree $d$ corresponds to a symmetric tensor of order $d$. Matrices are certainly easier to work with, more familiar, and more deeply understood, than higher-order tensors.
There is a theory of tensors. In fact it's a highly active area. But even so, it's leagues behind the theory of matrices.
In a little bit more detail: The symmetric $n \times n$ matrix $A$ with entries $a_{i,j} = a_{j,i}$ corresponds to the quadratic form
$$ Q(x) = \sum_{i,j} a_{i,j} x_i x_j = a_{1,1} x_1^2 + 2a_{1,2} x_1 x_2 + \dotsb = \vec{x}^T A \vec{x} , $$
and this is closely related to the bilinear form $B(x,y) = x^T A y$, or $\langle x | A | y\rangle$ if you like. When we go to cubic forms, the symmetric $n \times n \times n$ array $T$ with entries $a_{i,j,k} = a_{i,k,j} = a_{j,k,i} = \dotsb$ corresponds to the cubic form (degree $3$ homogeneous polynomial)
$$ C(x) = \sum_{i,j,k} a_{i,j,k} x_i x_j x_k = a_{1,1,1} x_1^3 + 3a_{1,1,2} x_1^2 x_2 + \dotsb + 6 a_{1,2,3} x_1 x_2 x_3 + \dotsb. $$
And it corresponds to the trilinear form $F(x,y,z)$ given by triple contraction of $T$ by the vectors $x,y,z$ along the three "sides" of $T$. How often do we deal with trilinear forms? They exist, but it's a bit of a minor topic compared with bilinear forms. Worse yet, $d$-linear forms for larger $d$.
I'm not claiming that higher-degree forms are lacking theory (or harder, or less interesting) because of the distinction between tensors and matrices. The relation here is probably more like an analogy or parallel, rather than causation.
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