Dynamic matrix rank with partial lookahead

We consider the problem of maintaining information about the rank of a matrix $M$ under changes to its entries. For an $n times n$ matrix $M$, we show an amortized upper bound of $O(n^{omega-1})$ arithmetic operations per change for this problem, where $omega < 2.376$ is the exponent for matrix multiplication, under the assumption that there is a {em lookahead} of up to $Theta(n)$ locations. That is, we know up to the next $Theta(n)$ locations $(i_1,j_1),(i_2,j_2),ldots,$ whose entries are going to change, in advance; however we do not know the new entries in these locations in advance. We get the new entries in these locations in a dynamic manner