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 maxrankS (M) = max(a1 ,a2 ,...at ) Î St rank M(a1 ,a2 ,...at )\max rank_S (M) = \mathop {max}\limits_{(a_1 ,a_2 ,...a_t ) \in S^t } rank M(a_1 ,a_2 ,...a_t )
and
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minrankS (M) = min(a1 ,a2 ,...at ) Î St rank M(a1 ,a2 ,...at ).\min rank_S (M) = \mathop {min}\limits_{(a_1 ,a_2 ,...a_t ) \in S^t } rank M(a_1 ,a_2 ,...a_t ).
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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 PSPACE-solvable to unsolvable.