In 1955, Serre conjectured that every row vector with entries in a commutative polynomial ring R=k[x_1, …, x_n] over a field k, admitting a right inverse over R, could be completed into a square matrix whose determinant is 1. That conjecture was independently proved by Quillen and Suslin in 1976 and is nowadays called the Quillen-Suslin theorem. Within module theory, the Quillen-Suslin theorem asserts that every finitely generated projective R-module is free and the computation of a completion is equivalent to the computation of a basis of the free module.

Different effective proofs of the Quillen-Suslin theorem have been given in the literature.

The first purpose of this talk is to explain the main ideas of an effective proof that follows a work of Logar and Sturmfels. We shall then demonstrate its implementation in the Maple package QuillenSuslin. Finally, we shall show different applications of the Quillen-Suslin theorem to mathematical systems theory (e.g., computation of flat outputs of functional linear systems, (Serre) reduction and decomposition problems, Lin-Bose’s conjectures, computation of (weakly) doubly coprime factorizations of multidimensional systems).

All the main results and algorithms will be illustrated by explicit examples.

This work was done in collaboration with my former PhD student Anna Fabianska.