Why are quadratic forms so special and why not investigate in higher forms?

Question

Ok, this is a soft question.

If $K$ is a field of characteristic different from $2$, one can use the polarization identity to get a one-to-one correspondence between

• homogeneous polynomials of degree $2$ in $K[X_1,...X_n]$ variables,
• symmetric bilinear forms $K^n\times K^n\to K$,
• symmetric $n\times n$-matrices with $K$-entries.

Quadratic forms are extensively studied. Why not to study 'cubic forms' or more generally '$n$-forms' in the same intensity? Perhaps one gets a correspondence of cubic forms with $3$-dimensional $n\times n\times n$-matrices. Is it just not investigated so much as quadratic forms just because these 'higer dimensional matrices' are more difficult to handle?

Answer 1

Because quadratic anything is usually easier that the same thing with higher powers.

Look at quadratic reciprocity compared with higher order reciprocity.

As a most elementary example, look at quadratic equations.

As a less elementary example, how about Fermat's Last Theorem.

Of course, this answer may reflect my mathematical ignorance. As the saying goes, to my family, I'm a mathematician, but to a mathematician, I'm no mathematician.

Answer 2

The recent work of Bhargava and his collaborators is (among other things) aimed at studying higher degree forms, higher dimensional matrices, and so on. You might be interested in looking at it, if only to get an idea of what people are doing. (If you search for Manjul Bhargava on the arxiv, you will find lots of papers.)