Constructions in R[x1,…,xn]: applications to K-theory

A classical result in K-theory about polynomial rings like the Quillen–Suslin theorem admits an algorithmic approach when the ring of coefficients has some computational properties, associated with Gröbner bases. There are several algorithms when we work in image, image a field. In this paper we compute a free basis of a finitely generated projective module over R[x1,…,xn], R a principal ideal domain with additional properties, test the freeness for projective modules over D[x1,…,xn], with D a Dedekind domain like image and for the one variable case compute a free basis if there exists any.