Observations concerning the complexity of a class of on-line algebraic problems

Defines and studies a class of on-line algebraic problems; a particular problem in this class is specified by providing an n×n matrix A. The author shows that the question of algebraic complexity reduces to determining the existence of what he calls (r, u) factorizations of A. Given an n×n matrix A, a factorization A =RU (R and U are n×m and m×n matrices, respectively; m unconstrained) is called an (r, u) factorization, provided that no row of R has more than r nonzero entries and no column of U has more than u nonzero entries. The existence of (r, u) factorization is explored from a general perspective.