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

We define the multidimensional unwrapped phase for any finite extent multidimensional signal that may have its zero on the distinguished boundary of the unit polydisc. By using this definition, we deduce that multivariate stable polynomials can be simply characterized in terms of the proposed unwrapped phase. A rigorous symbolic algebraic solution to the exact phase unwrapping problem for multidimensional finite extent signals is also proposed. This solution is based on a newly developed general Sturm sequence and does not need any numerical root finding or numerical integration technique. Furthermore, it is shown that the proposed algebraic phase unwrapping algorithm can be used to determine the exact zero distribution of any univariate complex polynomial without suffering the so-called singular case problem