Computation of a GCRD of several polynomial matrices using Trimmed Sylvester matrices

Abstract

This paper proposes a method for computing an exact Greatest Common Right Divisor (GCRD) of several univariate polynomial matrices using Trimmed Sylvester matrices. It establishes a novel relationship: for a full rank polynomial matrix with specified column degrees, constructed by stacking the given set of polynomial matrices one below the other, the degree of the determinant of any GCRD is related to the rank deficiency of a particular Trimmed Sylvester matrix. It characterizes the nullity of Trimmed Sylvester matrices contrasting it with the nullity of the generalized Sylvester matrices, proving that both of them are governed by different forms of the Index Sum Theorem. The coefficients of a GCRD are extracted using an Effectively Eliminating $QR$ ($EEQR$) decomposition of the Trimmed Sylvester matrix. The effectiveness of this approach is demonstrated through numerical examples, which reveal that the Trimmed Sylvester matrix approach offers a computationally advantageous alternative to the standard generalized Sylvester matrices, particularly when column degrees vary a lot.