The computational complexity of some problems of linear algebra (extended abstract)

In this paper we consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. Let x(1),x(2),...,x(t) be variables. Given a matrix M = M(x(1),x(2),...,x(t)) with entries chosen from E boolean OR {x(1),x(2),...,x(t)}, we want to determine maxrank(s)(M) = max/(a1,a2,...,at)is an element of st rank M(a(1),a(2),...,a(t)) and minrank(s)(M) = min/(a(1),a(2),...,a(t))is an element of st rank M(a(1),a(2),...,a(t)). There are also variants of these problems that specify more about the structure of M, or instead of asking for the minimum or maximum rank, ask if there is some substitution of the variables that makes the matrix invertible or noninvertible. Depending on E,S, and on which variant is studied, the complexity of these problems can range from polynomial-time solvable to random polynomial-time solvable to NP-complete to PSPA CE-solvable to unsolvable.