A *polymatroid* is a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that

1) $d(\varnothing)=0$,

2) $A \subset B$ implies $d(A) \leq d(B)$, and

3) $d(A \cap B) + d(A \cup B) \leq d(A) + d(B)$ for all $A,B \in P(X)$.

A polymatroid is said to be *representable* over $GF(2)$ (the field with two elements), if there exists a collection of subvectorspaces $\lbrace V_x \mid x \in X \rbrace$ of $GF(2)^{\oplus d(X)}$, such that

$$d(A) = \dim_{GF(2)} \bigvee_{x \in A} V_x, \quad \forall A \in P(X).$$

(It is clear that any function $d$, which is defined this way is indeed a polymatroid.)

A polymatroid is called *matroid* if $d(\lbrace x\rbrace)=1$ for all $x \in X$.
Tutte proved that a matroid is representable over $GF(2)$ if and only if the matroid $U_{2,4}$ does not appear as a minor. Here, $U_{2,4}$ is the matroid formed by four points that lie on one line, i.e. the underlying set is $\lbrace x_1,\dots,x_4 \rbrace$ and $d(\lbrace x_i,x_j\rbrace) = 2$ for $i \neq j$ and $d(\lbrace x_1,\dots,x_4\rbrace)=2$. (The necessity of this condition is obvious, since $GF(2)^{\oplus 2}$ has only $3$ non-zero elements.)

Question:Is there any useful characterization of the representability of a polymatroid over $GF(2)$ ? What is known about this question ?

minor. Note that a matroid may have a 4-point line as a minor, but not as a restriction, e.g. $U_{3,5}$. – Tony Huynh Sep 2 '10 at 22:54