Reidemeister Torsion, Peripheral Complex, and Alexander Polynomials of Hypersurface Complements

Let $f:\text{CN}\to \text{C}$ be a polynomial, which is transversal (or regular) at infinity. Let $\text{U}=\text{CN}\setminus f^{-1}(0)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give several estimates for the (infinite cyclic) Alexander polynomials of $\text{U}$ induced by $f$, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of $\text{U}$.