Polynomial description of inhomogeneous topological superconducting wires

We present universal features of the topological invariant of p-wave superconducting wires after the inclusion of spatial inhomogeneities. Three classes of distributed potentials are studied, a single- impurity, a commensurate and an incommensurate model using periodic site modulations. An analytical polynomial description is achieved by splitting the topological invariant into two parts, one dependent on the chemical potential and the other not. For the homogeneous case, an elliptical region is found where the topological invariant oscillates. The zeros of these oscillations occur at points where the fermion parity switches for finite wires. The increase of these oscillations with the inhomogeneity strength leads to new non-topological phases. We characterize these new phases according to each class of spatial distributions. Such phases could also be observed in the XY model.