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What are the subsets $V\subseteq\mathbb{Z}^n$ such that there is an integer combination of vectors in $V$ equal to $(1, 1, 1, \ldots)$? (where $n \in \mathbb{Z}^+$ and $\mathbb{Z}^n$ is the n-ary Cartesian product over $\mathbb{Z}$)

By integer combination I mean a positive integer $p$ along with vectors $v_1,v_2,\ldots,v_p \in V$ and integers $z_1,z_2,\ldots,z_p$ such that $z_1v_1 + z_2v_2 + \cdots + z_pv_p = (1,1,1,\ldots)$ with componentwise multiplication and addition.

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    Well, thanks anyway all.2012-05-06

1 Answers 1

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Let's do another very small special case to get some idea of what the problem is. Take $n=2$ and $V$ of cardinality 2, so $V=\{{(a,b),(c,d)\}}$ and we want to know whether there are integers $x,y$ such that $\pmatrix{a&c\cr b&d\cr}\pmatrix{x\cr y\cr}=\pmatrix{1\cr1\cr}$ By Cramer's Rule, this happens if and only if $a-b$ and $c-d$ are both multiples of $ad-bc$.