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Báo cáo toán học: "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"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: 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. | 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 Dan Singer Department of Mathematics and Statistics Minnesota State University Mankato dan.singer@mnsu.edu Submitted Dec 10 2008 Accepted Mar 23 2009 Published Mar 31 2009 Mathematics Subject Classifications 05C99 05E99 14R15 15A03 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 X n matrix containing an n X 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 finitedimensional 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 M 3 xi m 0 for m 3 M 4 3 m 0 for 5 m 8 and M 4 4 s 0. The first two properties yield combinatorial proofs of special cases of the homogeneous symmetric reduction of the Jacobian conjecture. 1 Introduction A classic problem in algebraic combinatorics is to show that the ring of symmetric functions in n variables An Z xi . xn Sn is generated by the elementary symmetric functions e1 . en and that the latter are algebraically independent over Z. The proof as given in 8 is to define e e 1 e 2 for each descending partition A A1 A2 . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R43 1 with parts of size n then observe that ex mx 22 axMmM M .