Singular, nonsingular, and bounded rank completions of ACI-matrices

An affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns. A completion is a matrix obtained by giving values to each of the indeterminates. Affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate. We determine when the rank of all completions of an affine column independent matrix is bounded by a given number, generalizing known results for partial matrices. We also characterize the square partial matrices over a field all of whose completions are nonsingular. The maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries.