Higher Order Independence in Matroids

One may regard vectors in a finite dimensional vector space as being linear forms in a polynomial ring in an obvious way. A collection of linear forms satisfying various linear dependence relations can become independent when each of the forms is raised to the k-th power. In this paper we prove that a certain class of matroids satisfies a " higher order " independence property of this kind. The case k = 2 is of particular importance, and we mention a number of applications to topology, algebraic geometry, electrical networks and chemical kinetics. 1. Power independence We assume that the reader is familiar with the terminology and elementary theory of matroids (combinatorial geometries) as found in, for example, Crapo-Rota [5]. We say that a matroid is simple if the empty set as well as every point is closed. If we can coordinatize a matroid with the columns of a nonzero matrix A, then the matroid is simple if and only if no column of A is a multiple of another. If G(S) is a nonsimple matroid, its simplification is the simple matroid obtained by removing the points in 0 and identifying all multiple points of G(S). Let V be the vector space K" of n x 1 column vectors with coordinates in a field K. We write S(V) for the A>th symmetric power of V and V® for the A:-th tensor power of V. The vector space V is isomorphic to the space of linear forms in the indeterminates Xx, X2, ••-, Xn; and, if we identify V with this space, then S (V) is the space of homogeneous polynomials of degree k in the variables Xu X2, •-., Xn. Let A be an n x s matrix with entries in K. We write G(A) for the matroid whose points are the columns of A and whose closure operation is the usual linear span of (column) vectors. The A>th symmetric powers of the columns of A are a set of vectors in S(V), whose associated matroid will be denoted Gk(A). More concretely, let the entries of A be atj. The columns of A may be regarded as being the linear forms {(E««o-^i)U ^ J'^ )> nd the k-th symmetric powers of the columns of A correspond to the polynomials {(Zi#jj-X',)'|l O ^ s}. Then Gk(A) is the matroid associated to this set of elements of S(V). Similarly, we write G®k(A) for the matroid associated to the set of k-th tensor powers of the columns of A. It is easy to check that Gk(A) is isomorphic to G®k(A) when char(X) is zero or larger than k but that in general they are not isomorphic. It is also easy to give an example which shows that the matroids Gk(A) and Gm(A) are not determined by the matroid G(A). Received 14 April, 1977; revised 11 October, 1977 and 12 July, 1978. The first author was supported by NSF #MCS 77-01947; the second author was supported by NSF # MCS 76-O5823A01. [J. LONDON MATH. SOC. (2), 19 (1979), 193-202] 194 KENNETH BACLA.WSKI AND NEIL L. WHITE Example 1. Consider the following two matrices: