Abstract:
We consider the computational complexity of some problems
dealing with matrix rank. Let E, S be subsets of a commutative ring R.
Let 
be variables. Given a matrix 
with entries chosen from 
, we want to determine
and
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.
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