An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n⩾3 and a unimodular vector V=t(v1,…,vn)∈Rn (that is, such that 〈v1,…,vn〉=R), the algorithm computes a matrix M in Mn(R) whose determinant is a monomial such that MV=t(1,0,…,0), and thus M-1 is a completion of V to an invertible matrix.