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A graph-theoretic approach to a conjecture of Dixon and Pressman Brassil, Matthew


Given n×n matrices, A_1,...,A_k, define the linear operator L(A_1,...,A_k): Mat_n -> Mat_n by L(A_1,...,A_k)(A_(k+1)) = sum_sigma sgn(sigma) sgn(sigma)A_sigma(1)A_sigma(2)...A_sigma(k+1). The Amitsur-Levitzki theorem asserts that L(A_1,...,A_k) is identically 0 for every k > 2n − 1. Dixon and Pressman conjectured that if 2 <= k <= 2n − 2, then for A_1,...,A_k ∈ Mat_n(R) in general position, the kernel of L(A_1,...,A_k) has dimension k when k is even and either k +1 or k + 2 when k is odd (depending on whether n is even or odd). We prove this conjecture in the case where k is even. Our proof relies on graph-theoretic techniques.

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