Algebraic multidimensional phase unwrapping and zero distribution of complex polynomials -- characterization of multivariate stable polynomials (Q2724230)

The paper provides an algorithm for multidimensional phase unwrapping, a problem with applications in signal processing. The algorithm, described in section IV, is purely algebraic, assuming infinite precision arithmetic and using symbolic computations. It is based on a generalization of Sturm sequences and does not rely on polynomial root finding or numerical integration. The algorithm can compute the unwrapped phase of a multivariate polynomial or signal at any frequency point. The second contribution of the paper is in investigating relations between the phase unwrapping problem and zero distribution of multivariate polynomials. It is namely shown in Theorem 3 that stability of a multivariate polynomial is equivalent to periodicity of its unwrapped phase. In the case of one- or two-indeterminate polynomials, the phase unwrapping algorithm therefore allows one to visualize stability conditions. The paper certainly contains interesting results, but in my opinion it is very long with respect to its contribution and not especially well-written. Notations are sometimes awkward, and some sections are quite technical and difficult to follow.