  Let $K$ be an infinite field and $R=K[x_1,...,x_n]$ be the polynomial ring.
Let $V=V_1, ..., V_m$ be a collection of vector spaces of linear forms. Denote
by $A(V)$ the $K$-subalgebra of $R$ generated by the elements of the product
$V_1... V_m$. Our goal is to investigate the properties of the algebra $A(V)$
and the relations with two problems in algebraic combinatorics White's and
related conjectures on polymatroids and the study of integral posets.
