# r-locally linear functions of many variables

Suppose $m,n,r$ are positive integers. Suppose V is a $m$-dimensional vector space over a field F. Let $G(V,n,r)$ denote the space of $n$-tuples of elements of V with the property that the vector space generated by any such tuple has dimension at most r. In other words:

$$G(V,n,r) = \bigcup_{W \subseteq V, \operatorname{dim}(W) \le r} W^n$$

Let $S$ be the space of functions $f:G(V,n,r) \to F$ with the property that for any subspace W of $V$ of dimension at most r, the restriction of f to $W^n$ is a multilinear $n$-form on $W$.

Clearly, $S$ is a vector space under pointwise addition and scalar multiplication of functions. Question: What is the dimension of $S$ as a function of $m$, $n$, and $r$ (and if necessary, assume the base field is finite of size $q$)? Assume $r \le m$ and $r \le n$ (otherwise the problem becomes trivial)? Also, is there a nice way of thinking of $S$ as linear functionals on some vector space?

Special cases:

1. When $r \ge \min \{ m, n \}$, every $n$-tuple of elements of $V$ lies inside a $r$-dimensional subspace, so $G(V,n,r) = V^n$, and the condition of being a function as described above just translates to being a linear functional on the $n^{th}$ tensor power $V^{\otimes n}$. The dimension of this vector space is $m^n$.
2. When $r = 1$, then $G(V,n,r)$ comprises the tuples where all the entries are either zero or are all multiples of a single vector. The linearity condition in this case simplifies to just saying that we get an arbitrary function on the projective space of $V$. The dimension of this vector space depends on the size of the base field, and it is the same as the cardinality of the projective space. In particular, if the base field is infinite, this space would be infinite-dimensional.

Motivation: I'm interested in considering functions that are like linear forms, except that they need to behave that way only when the inputs come from a certain small-dimensional subspace. In particular, the case $r = 3$ interests me, because most algebraic identities are quantified with at most three variables at a time.

• "linear $n$-form" means "MULTIlinear $n$-form", right? – darij grinberg Apr 5 '11 at 23:33
• Darij: Yes, you're right. I fixed it. – Vipul Naik Apr 6 '11 at 16:06