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?

  • $\begingroup$ I'm not really sure but it may have something to do with $L^2$ being the only Hilbert space among the $L^p$ spaces. $\endgroup$
    – tomcuchta
    Oct 20, 2013 at 23:23
  • $\begingroup$ see math.stackexchange.com/questions/329936/… $\endgroup$
    – Will Jagy
    Oct 21, 2013 at 0:25
  • $\begingroup$ You might be interested to know that there's a book by Y. Manin entitled Cubic Forms. $\endgroup$
    – user64687
    Oct 30, 2013 at 21:48
  • $\begingroup$ A related Mathoverflow question: mathoverflow.net/questions/430365/… $\endgroup$
    – KCd
    Dec 26, 2023 at 20:39

2 Answers 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.)


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.


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