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{\bf Dan Singer}
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{\bf 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}
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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.

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