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 form E union {x_1,x_2,...,x_t}, we want to determine 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 polyonial-time solvable to random ploynomial-time solvable to NP-complete to PSPACE-solvable to unsolvable.
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机译：在本文中，我们考虑了处理矩阵等级的一些问题的计算复杂性。让e，s是换向环R的子集。设X_1，X_2，...，X_T是变量。给定矩阵m = m（x_1，x_2，...，x_t）所选择的表单e Union {x_1，x_2，...，x_t}，我们想确定这些问题的变体还可以指定更多关于M的结构，或者不是询问最小或最大等级，询问是否存在使矩阵可逆或不可逆转的变量的一些替代。根据E，S和在哪些变体的研究，这些问题的复杂性可以从多龙 - 时间可溶解到随机繁殖的时间溶解到NP-Completable至Pspace-overvable至无法解决的。
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