Unlabeled equivalence for matroids representable over finite fields

Two r × n matrices A and A′ representing the same matroid M over GF (q), where q is a prime power, are projective equivalent representations of M if one can be obtained from the other by elementary row operations and column scaling. Bounds for projective inequivalence are difficult to obtain and are known for only a few special classes of matroids. In this paper we define two matrices A and A′ to be geometric equivalent if, in addition to row operations and column scaling, column permutations are also allowed. We show that the number of geometric inequivalent representations is at most the number of projective inequivalent representions and we give a polynomial time algorithm for determining if two projective inequivalent representations are geometrically equivalent. Thus, from a computational perspective there is no additional cost to altering the definition of equivalence in this manner. The benefit is that it could lead to a new set of theorems for inequivalence with better bounds.