Unimodular completion of polynomial matrices

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

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