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Given a nonzero linear differential polynomial $A$ in $\\id$ we give necessary and sufficient conditions on $A$ for $\\cP$ to be $n-1$ dimensional. We prove the existence of a linear perturbation $\\cP_{\\phi}$ of $\\cP$ so that the linear complete differential resultant $\\dcres_{\\phi}$ associated to $\\cP_{\\phi}$ is nonzero. 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