TITLE: Is there a rank for higher degree homogeneous forms analogous to that of quadratic forms?
QUESTION [8 upvotes]: Given a quadratic form $Q(x_1, ..., x_n)$, there is a natural notion of rank defined by looking at the rank of the unique symmetric matrix associated to the quadratic form, i.e. we consider the symmetric matrix $A$ such that
$\mathbf{x}^T A \mathbf{x} = Q(\mathbf{x})$. 
Suppose we have a $F(\mathbf{x})$, a degree $d$ homogeneous form. I naively thought that perhaps the rank of $F$ should be defined to be the rank of a hypermatrix $A$ attached to $F(\mathbf{x})$ (Assuming such notions makes sense). But I have never seen anything like that in the literature. 
So perhaps there are issues if one wants to define a rank of forms this way. 
I was wondering if someone could explain me why considering this may not make sense or not considered? Thank you very much!

REPLY [8 votes]: When $F$ is a degree $d$ homogeneous form on $n$ variables in a finite field ${\bf F}_q$, in the regime when $n$ is large and the order of the field and the degree is small, there are two related notions of rank that are useful.  One is an "algebraic" notion of rank - $F$ has rank at most $r$ if $F(x)$ can be expressed as $F(x) = f( Q_1(x),\dots,Q_r(x))$ for some forms $Q_1,\dots,Q_r$ of degree strictly less than $d$, and some function $f$.  (For comparison, note that a rank $r$ quadratic form $F(x)$ can be written in terms of $r$ linear functions of $x$.)  See for instance this paper of Ben Green and myself where this notion is used to control the equidistribution of $F$.
A related notion is the "analytic" notion of rank introduced by Gowers and Wolf - a form $F$ has analytic rank $r$ if the exponential sum $\frac{1}{q^n} \sum_{x \in {\bf F}_q^n} \chi( F(x) )$ has magnitude $q^{-r/2}$, where $\chi$ is some fixed character on ${\bf F}_q$.  For quadratic forms in odd characteristic this matches the usual notion of rank by the familiar properties of Gauss sums.