Constructions in R[x_1, ..., x_n]. Applications to K-Theory

Abstract

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 $\K[\x]$, $\K$ a field. In this paper we compute a free basis of a finitely generated projective module over $R[\x]$, $R$ a principal ideal domain with additional properties, test the freeness for projective modules over $D[\x]$, with $D$ a Dedekind domain like $\Zset[\sqrt{-5}]$ and for the one variable case compute a free basis if there exists any.