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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!

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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.

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Thank you very much for your answer. Does the algebraic rank you mentioned ever been used for forms over $\mathbb{C}, \mathbb{R}, \mathbb{Q}$ before? Also if we define algebraic rank analogously for forms over these fields, would you also expect it could control equidistribution? Thanks! – Johnny T. Mar 19 at 16:00
I don't know of any such use in the literature. It's also not clear what equidistribution would even mean when the range of the polynomial is non-compact - one may need to do some sort of "resolution of singularities" of the polynomial at infinity before one has a meaningful question to ask here, – Terry Tao Mar 22 at 15:44

The most basic notion of rank for a homogeneous symmetric $d$-form $F$ is the dimension $k$ of the span of the partials $$ F_i = \frac{\partial F}{\partial x^i} $$ in the space of polynomials of degree $d{-}1$. One always has $k\le n$, and it is always possible to write $F$ as a polynomial in $k$ variables (and no fewer). This is an elementary result.

There are other notions, such as the monomial rank, i.e., the least number of monomial terms needed to express $F$ with respect to some basis of the variables, which need bear no relation with $n$ (other than being bounded above by a constant depending only on $d$ and $n$). There is also the power rank, which is the least number of terms needed to express $F$ as linear combination of pure $d$-th powers. As far as I know, neither of these is easily computable when $d>2$ and $n$ is not small.

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Thank you very much for your answer. I was wondering if you could clarify something: we have a symmetric form $F$ in $n$ variables, and then you say it can be written as a polynomial in $k$ variables. I am confused with what you meant by this. Thanks! – Johnny T. Mar 19 at 16:06
@JohnnyT.: What I meant was, for example, this: The quadratic form $x^2 + 2xy + y^2$ uses two variables, but it is clearly $z^2$ where $z = x+y$, so by a change of variables, it can be written as a quadratic form in one variable, namely, $z$. – Robert Bryant Mar 19 at 16:50

Such notions are defined for tensors. A tensor can be seen as you said like, a hyper matrix $A\in\Bbb R^{n_1\times \ldots \times n_d}$ with $n_1,\ldots,n_d\in\Bbb N$. Where $A$ has $d$ indices instead of $2$, i.e. $A_{j_1,\ldots,j_d}\in\Bbb R$ for $j_i=1,\ldots,n_i$ and $i=1,\ldots,d$. There are several notions of rank defined for tensors. You have the multilinear rank (based on one generalization of the singular value decomposition), the rank given by the canonical polyadique decomposition (another generalization of the SVD), the border rank and the cactus rank (more abstract concepts). There are other types of decompositions ($\approx$ generalizations of the usual spectral decompositions) which somehow induce other possible notions of rank.

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