Undecidability of c-Arrangement Matroid Representations

A c-arrangement is an arrangement of dimension-c subspaces such that all their sums have dimension a multiple of c. Matroids arising as normalized rank functions of c-arrangements are also known as multilinear matroids. We prove that there is no algorithm to decide whether a matroid is multilinear. In particular there is no algorithm to decide whether there exists a representable multiple of a given polymatroid.