Contents I. Introduction
The algorithmic algebra of Gröbner bases was developed by Buchberger in the 1960s and further enriched with contributions from him and many other researchers (see, e.g., [15]–[19], [55], and the references therein). Gröbner bases theory and its predecessors (like standard bases of Hironaka [50]) provide a rich theoretical framework in algebraic geometry and commutative algebra, particularly polynomial ideal theory. The algorithmic algebra of Gröbner bases has wide-ranging applications in theoretical physics, applied science, and engineering. The main reason for the success of Gröbner bases is that many problems in mathematics, science, and engineering can be represented by multivariate polynomials (e.g., ideals, modules, and matrices), where Gröbner bases play a role similar to the role of Euclidean Division Algorithm in the Euclidean ring of univariate polynomials. In 2001, a special issue on the applications of Gröbner bases to multidimensional systems and signal processing, published in Multidimensional Systems and Signal Processing, was guest-edited by two of the authors to emphasize the useful role of Gröbner bases to researchers in the area of multidimensional (-D) systems theory and signal processing [65]. A special issue on multidimensional signals and systems, published in this journal [5], substantiated the increasing scope of the subject-matter. The present paper intends to respond to the accelerated need of Gröbner bases to these momentous developments and to attract new researchers with new applications that may prove to be a fertile ground for implementing the developments in algorithmic polynomial ideal theory.
