In this note we prove the following surprising characterization: if $X\,\subset \,{{\mathbb{A}}^{n}}$ is an (embedded, non-empty, proper) algebraic variety deûned over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ of logarithmic vector fields of $X$ is a reflexive ${{O}_{{{\mathbb{A}}^{n}}}}$-module. As a consequence of this result, we derive that if ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ is a free ${{O}_{{{\mathbb{A}}^{n}}}}$-module, which is shown to be equivalent to the freeness of the $t$-th exterior power of ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ for some (in fact, any) $t\,\le \,n$, then necessarily $X$ is a Saito free divisor.

In this article, we prove that a free divisor in a three-dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. Calderón-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem that describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.