Further results on Serre’s reduction of multidimensional linear systems

Serre’s reduction aims at reducing the number of unknowns and equations of a linear functional system (e.g., system of ordinary or partial differential equations, system of differential time-delay equations, system of difference equations). Finding an equivalent representation of a linear functional system containing fewer equations and fewer unknowns generally simplifies the study of its structural properties, its closed-form integration and different numerical issues. The purpose of this paper is to present a constructive approach to Serre’s reduction for linear functional systems. I. AN ALGEBRAIC ANALYSIS APPROACH TO LINEAR SYSTEMS THEORY In what follows, D will denote a noncommutative noetherian domain, namely, a unital ring satisfying that d d′ is not necessarily equal to d′ d for d, d′ ∈ D, containing no nontrivial zero-divisors, i.e., d d′ = 0 yields d = 0 or d′ = 0, and every left (resp., right) ideal of D is finitely generated, i.e., can be generated by a finite family of elements of D as a left (resp., right) D-module ([9], [16]). Moreover, we shall denote by D1×p (resp., D) the left (resp., right) D-module formed by row (resp., column) vectors of length p (resp., q) with entries in D and by R ∈ Dq×p a q × p matrix R with entries in D. Moreover, we shall use the following notations: .R : D1×q −→ D1×p μ 7−→ μR, R. : D −→ D η 7−→ Rη. (1) Since the image imD(.R) = D1×q R of the left Dhomomorphism .R : D1×q −→ D1×p defined by (1), i.e., imD(.R) = {λ ∈ D1×p | ∃ μ ∈ D1×q : λ = μR}, is a left D-submodule of D1×p, we can introduce the quotient left D-module M = D1×p/(D1×q R) and the left Dhomomorphism π : D1×p −→ M which sends λ ∈ D1×p to its residue class π(λ) in M . In particular, π(λ) = π(λ′) iff there exists μ ∈ D1×q such that λ − λ′ = μR. The left D-module M = D1×p/(D1×q R) is then said to be finitely presented by R ([16]). Let us describe the left D-module M = D1×p/(D1×q R) in terms of generators and relations. Let {fj}j=1,...,p be the standard basis of the left D-module D1×p, namely, fj is the row vector of length p with 1 at the jth position and 0 elsewhere, and yj , π(fj) ∈ M for j = 1, . . . , p. Since every m ∈ M has the form m = π(λ) for a certain row vector λ = (λ1 . . . λp) ∈ D1×p, m = π  p ∑