A Graph-Theoretic Method for Choosing a Spanning Set for a Finite-Dimensional Vector Space, with Applications to the Grossman-Larson-Wright Module and the Jacobian Conjecture

Abstract

It is well known that a square zero pattern matrix guarantees non-singularity if and only if it is permutationally equivalent to a triangular pattern with nonzero diagonal entries. It is also well known that a nonnegative square pattern matrix with positive main diagonal is sign nonsingular if and only if its associated digraph does not have any directed cycles of even length. Any $m\times n$ matrix containing an $n\times n$ sub-matrix with either of these forms will have full rank. We translate this idea into a graph-theoretic method for finding a spanning set of vectors for a finite-dimensional vector space from among a set of vectors generated combinatorially. This method is particularly useful when there is no convenient ordering of vectors and no upper bound to the dimensions of the vector spaces we are dealing with. We use our method to prove three properties of the Grossman-Larson-Wright module originally described by David Wright: $\overline{\cal M}(3,\infty)_m=0$ for $m\ge 3$, $\overline{\cal M}(4,3)_m=0$ for $5\le m\le 8$, and $\overline{\cal M}(4,4)_8=0$. The first two properties yield combinatorial proofs of special cases of the homogeneous symmetric reduction of the Jacobian conjecture.