Polynomial matrix primitive factorization over arbitrary coefficient field and related results

Factorizations fornD polynomial matrices

“…In this paper, when we state an nD polynomial matrix [7], [10]- [19], [23], [24], [28], [32], [33]. However, because an nD polynomial equation g(z) = 0 could also mean solving g(z) ----0 for some (but not all) (z) ~ C n, in the following lemma and its proof, we shall use the expression g(z) =-0 for the case when g(z) ---0 for every (z) ~ C n. In the rest of the paper, we shall still use the conventional symbol "=" instead of "-~" when such a usage does not cause any confusion.…”

“…The problems of multivariate (nD) polynomial factorizations and nD polynomial matrix factorizations have attracted much attention over the past decades because of their wide applications in multidimensional (nD) circuits, systems, controls, signal processing, and other areas (see, e.g., [1]- [7], [10]- [19], [21]- [24], [28], [30], [32], [33]). For an arbitrary nD polynomial over the field of real numbers or the field of complex numbers, although the existence of its factorization into a product of irreducible polynomials has been known for a long time (see, e.g., [2]), constructive algorithms for carrying out such a factorization are available only for some classes of nD polynomials (see, e.g., [21], [22], [30]).…”

“…For example, it is still unknown [32], [10], [19], [12, p. 63] how to extract a common factor from two nD (n > 2) polynomial matrices. In fact, this open problem is closely related to the factorization of a normal full rank nD polynomial matrix into a product of two nD polynomial matrices, with one of them being prime in some sense [23], [32], [7].…”

“…In fact, this open problem is closely related to the factorization of a normal full rank nD polynomial matrix into a product of two nD polynomial matrices, with one of them being prime in some sense [23], [32], [7]. This prime factorization problem has long been solved for one-dimensional (1 D) and two-dimensional (2D) polynomial matrices [23], [10], [11]. However, it is a challenging open problem for nD (n > 2) polynomial matrices [32], [7], [12, p. 63] because of some fundamental differences between nD polynomial matrices and their 1D and, 2D counterparts [32], [14], [7], [12, p. 63].…”

“…Charoenlarpnopparut Another related matrix factorization problem is the determinantal factorization problem for a given nD (n _> 1) polynomial matrix: to extract a factor with determinant equal to a divisor of the determinant of the given matrix. Again, it is well known [23], [10] that any 1D and 2D polynomial matrices admit determinantal factorizations. However, it has been pointed out [32], [ l 3] that some nD (n > 2) polynomial matrices do not have determinantal factorizations.…”