Unimodular completion of polynomial matrices

Abstract

Given a rectangular matrix F ∈ K[x]^mxn with m < n of univariate polynomials over a field K. we give an efficient algorithm for computing a unimodular completion of F. Our algorithm is deterministic and computes such a completion, when it exists, with cost O~ (n^ωs) field operations from K. Here s is the average of the m largest column degrees of F and ω is the exponent on the cost of matrix multiplication. Here O~ is big-O but with log factors removed. If a unimodular completion does not exist for F, our algorithm computes a unimodular completion for a right cofactor of a column basis of F, or equivalently, computes a completion that preserves the generalized determinant.