We study linear subspaces L ⊆ Mn (over an algebraically closed field F of characteristic zero) and their singular sets S(L) defined by S(L) = {A ∈ Mn : χ(A+ L) is not dense in Fn}, where χ : Mn −→ F n is the characteristic map. We give a complete characterization of the subspaces L ⊂ M2 such that ∅ 6= S(L) 6= M2. We also provide a complete characterization of the singular sets S(L) in the case ...

Let k be a field of characteristic zero and F : k → k a polynomial map of the form F = x + H, where H is homogeneous of degree d ≥ 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T−1HT = (0, h2(x1), h3(x1, x2)), where the hi are homogeneous of degree d. As a consequence of this r...