On the Computation of the GCD of 2-D Polynomials

An interesting problem of algebraic computation is the computation of the greatest common divisor (GCD) of a set of polynomials. The GCD is usually linked with the characterisation of zeros of a polynomial matrix description of a system. The problem of finding the GCD of a set of n polynomials on R[x] of a maximal degree q is a classical problem that has been considered before (Karcanias et al., 2004). Numerical methods for the GCD (Karcanias et al., 1994; Mitrouli el al., 1993) were also developed. Due to the difficulty in finding the exact GCD of a set of polynomials, approximate algorithms were also developed (Noda et al., 1991). A comparison of algorithms for the calculation of the GCD of polynomials is given in (Pace et al., 1973). The main disadvantage of many algorithms is their complexity. In order to overcome these difficulties, we may use other techniques such as interpolation methods. For example, Schuster et al. (1992) used interpolation techniques in order to find the inverse of a polynomial matrix. The speed of interpolation algorithms can be increased by using Discrete Fourier Transform (DFT) techniques or, better, Fast Fourier Transform (FFT) techniques. Some of the advantages of DFT based algorithms are that there are very efficient algorithms available in both software and hardware and that they are well suited for parallel environments (through symmetric multiprocessing or other techniques). Karampetakis et al. (2005) used DFT techniques to compute the minimal polynomial of a polynomial matrix, and Paccagnella et al. (1976) used FFT methods for the computation of the determinant of a polynomial matrix. Here we provide an algorithm for the computation of GCD of 2-D polynomials based on DFT techniques. The proposed algorithm is illustrated via examples.