The linear dependence problem for power linear maps

Let Bl, l = 1, … , k, be m × nl complex matrices and let be complex vector variables. We show that the components of the map H=(B1x[1])(d1)∘⋯∘(Bkx[k])(dk) are linearly dependent over C if and only if , where ∘ means the Hadamard product, X∗ and X(d) denote the conjugate transpose and the dth Hadamard power of a matrix X respectively. Connections are established between the Homogenous Dependence Problem (HDP(n,d)), which arises in the study of the Jacobian Conjecture, and the dependence problem for power linear maps (PLDP(n,d)). An algorithm is given to compute counterexamples to PLDP(n,d) from those to HDP(n,d), and counterexamples to PLDP(n,3) are obtained for all n⩾67.