Let \(D=k[x_1,\dots,x_n]\) with \(k\) a field. In the present paper, the authors consider multidimensional linear functional systems which are represented by a finitely generated \(D\)-module. For instance, \(D\) can be the ring of linear differential operators with rational coefficients \(\mathbb{Q}[\partial/\partial x_1,\partial/\partial x_2]\) or the ring of differential time-delay operators \(\mathbb{Q}[d/dt,\delta]\). The Quillen-Suslin theorem asserts that any projective \(D\)-module is free. The authors recall a constructive proof of that theorem and give four new applications to mathematical systems theory.
Their first motivation is the Monge problem, which consists in studying the injective parametrizations of a multidimensional functional system. For the systems considered in the paper, the existence of an injective parametrization is equivalent to the freeness of the \(D\)-module associated with the system [cf. F. Chyzak, A. Quadrat and D. Robertz, Appl. Algebra Eng. Commun. Comput. 16, No. 5, 319–376 (2005; Zbl 1109.93018)]. Using a Quillen-Suslin algorithm, in particular the construction of bases of free modules, the Monge problem is here constructively solved. As a corollary, it is proved that a multidimensional system is equivalent to a one-dimensional system if the module associated with the initial system is free.
Other new applications of the Quillen-Suslin theorem to mathematical system theory are also given: Lin-Boses’s conjecture; (weakly) left-/right-coprime factorizations of rational transfer matrices; decomposition of multidimensional linear functional systems.
The authors present detailed algorithms which are implemented in the package QUILLEN-SUSLIN (A. Fabiańska, QUILLENSUSLIN project: A package for computing bases of free modules over commutative polynomial rings, http://wwwb.math.rwth-aachen.de/QuillenSuslin), developed in the computer algebra system MAPLE. Numerous examples are given.
It should be emphasized that the paper is written in a pedagogical way; in particular many background notions are recalled.
For the entire collection see [Zbl 1130.93007].