  We present the first phase retrieval algorithm guaranteed to solve the
multidimensional phase retrieval problem in polynomial arithmetic complexity
without prior information. The method successfully terminates in O(N log(N))
operations for Fourier measurements with cardinality N. The algorithm is
guaranteed to succeed for a large class of objects, which we term "Schwarz
objects". We further present an easy-to-calculate and well-conditioned diagonal
operator that transforms any feasible phase-retrieval instance into one that is
solved by our method. We derive our method by combining techniques from
classical complex analysis, algebraic topology, and modern numerical analysis.
Concretely, we pose the phase retrieval problem as a multiplicative Cousin
problem, construct an approximate solution using a modified integral used for
the Schwarz problem, and refine the approximate solution to an exact solution
via standard optimization methods. We present numerical experimentation
demonstrating our algorithm's performance and its superiority to existing
method. Finally, we demonstrate that our method is robust against Gaussian
noise.
