  We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and
higher-order degrees $\delta_{n,i}(\mathcal{U})$ which are invariants of a
complex hypersurface complement $\mathcal{U}$. These invariants come from the
module structure of the homology of certain solvable covers of the hypersurface
complement. Such invariants were originally developed by T. Cochran in [1] and
S. Harvey in [8], and were used to study knots and 3-manifolds. In this paper,
I generalize the result proved by C. Leidy and L. Maxim [22] from the plane
curve complements to higher-dimensional hypersurface complements. Zariski
observed that the position of singularities on a singular complex plane curve
affects the topology of the curve. My results on higher-order degrees of
hypersurface complements also show that global topology is controlled by the
local topologies. In particular, the higher-order degrees of the hypersurface
complement are bounded by a linear combination of the higher-order degrees of
the local link pairs.
