Fibered cohomology classes in dimension three, twisted Alexander polynomials and Novikov homology

We prove that for "most" closed 3-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in H^1 (M,\bf R)$ is equivalent to the fact that every twisted Alexander polynomial $\Delta^H(M,u) \in {\bf Z}[G/\ker u]$ associated to a normal subgroup with finite index $H < \pi_1(M)$ has a unitary $u$-minimal term.
Comment: The statement of the main theorem has been corrected. This version has been accepted for publication in Annales de l'Institut Fourier