A general factorization method for multivariable polynomials

Abstract

The problem of factorizing a multivariable or multidimensional (m-D) polynomialf (z ₁,z ₂, ...,z _m ), with real or complex coefficients and independent variables, into a number of m-D polynomial factors that may involve any independent variable or combination of them is considered. The only restriction imposed is that all factors should be linear in one and the same variable (sayz ₁). This type of factorization is very near to the most general type:

$$\begin{gathered} f(z_1 ,z_2 , \ldots ,z_m ) = \prod\limits_{i = 1}^{N_1 } {\lbrack {\mathop \sum \limits_{\epsilon _1 = 0}^{\epsilon _{i,1} } \cdots \mathop \sum \limits_{\epsilon _m = 0}^{\epsilon _{i,m} } a_{i;\epsilon _1 ,\epsilon _2 , \ldots ,\epsilon _m } z_1^{\epsilon _1 } \cdots z_m^{\epsilon _m } + c_i } \rbrack} \hfill \\ (\epsilon _1 , \ldots ,\epsilon _m ) \neq (0, \ldots ,0) \hfill \\ \end{gathered}$$

and appears to be the most general type available. The method is first briefly sketched for the convenience of the reader, and then is presented in detailed form through a number of theorems. These theorems provide a clear algorithmic way for the factorization, which may be automated via a suitable computer code. The factorization of m-D polynomials simplifies the stability analysis and the realization of m-D systems, as well as the solution of distributed parameters systems.