五个或更多变量的线性秩不等式
Linear rank inequalities on five or more variables
摘要：
向量空间的子空间的秩满足所有由熵满足的线性不等式（包括标准的香农不等式）和一个由英格尔顿提出的附加不等式。众所周知，香农不等式和英格尔顿不等式生成的所有线性秩不等式最多可达四个变量，但它一直是一个悬而未决的问题，是否额外的不等式适用于五个或更多变量的情况。在这里，我们列出了24个不等式，连同香农和英格尔顿不等式，产生所有的线性秩不等式的五个变量。我们还给出了一个部分列表的线性秩不等式的六个变量和一般的结果，产生这样的不平等的任意数量的变量，我们证明，基本上有新的不等式在每个数量的变量超过4（结果也证明最近由Kinser）。
英文摘要：
Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether additional inequalities hold for the case of five or more variables. Here we give a list of 24 inequalities which, together with the Shannon and Ingleton inequalities, generate all linear rank inequalities on five variables. We also give a partial list of linear rank inequalities on six variables and general results which produce such inequalities on an arbitrary number of variables; we prove that there are essentially new inequalities at each number of variables beyond four (a result also proved recently by Kinser).
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