Timeline for Discrete disjoint covering of integer lattices

4 events

when toggle format	what		by	license	comment
May 9, 2013 at 18:26	comment	added	some guy on the street		I think that's what I was trying to say... but I'd have quibbled $\Sigma a_{ij} v_j$; and now I'm convinced that there are always exactly $n$ such $[a_{i\dot}]$; and so the interest is in how might this fail to give a basis for $\mathbb{Z}^n$... I'm now convinced this gives a list of all solutions in every case; I still like that Kevin's answer gives a specific solution in all dimensions, which is closer to what I was wondering about, but this is really nifty!
May 9, 2013 at 17:29	comment	added	Richard Stanley		For any $n\times n$ integer matrix $M$ with determinant $d\neq 0$ and columns $v_1,\dots,v_n$, there are exactly $\|d\|-1$ nonzero integer vectors $u_1,\dots,u_{d-1}$ of the form $u_i=\sum a_i v_i$, where $0\leq a_i<1$. The point here is that these nonzero $u_i$'s should form a basis for the lattice $\mathbb{Z}^n$.
May 9, 2013 at 15:34	comment	added	some guy on the street		could you explain the "$\Sigma a_i v_i=u_i"? I can see that there are finitely many HNFs to check, and that they're enough... are we basically trying to see if the cube spanned by an HNF contains enough independent integer points?
May 9, 2013 at 14:53	history	answered	Richard Stanley	CC BY-SA 3.0