My question arise at the consideration of Newton polytopes. In that context
I consider integer matrices $ A =(a_{ij})\in \mathbb{Z}^{(n+1) \times N} $ with $n+1 \leq N$ having the following special properties
- $a_{ij}\in \{0;1;2\}$
- first row is fixed: $a_{1j} = 1$ for all $j$
- the column-sums have only two possible values $\sum_{i=1}^{n+1} a_{ij} \in \{K;K+1\}$ with $2\leq K \leq n+1$
- columns which satisfy $\sum_{i=1}^{n+1} a_{ij} = K$ can only contain $a_{ij}\in\{0;1\}$
- all columns are distinct: for $j\neq j^\prime$ there is an $i$ with $a_{ij} \neq a_{ij^\prime}$
and I want to check if/prove that those matrices have the full rank: $\operatorname{rank} (A) = n+1$.
Also a proof for the special case $a_{ij}\in\{0;1\}$ would help me.
The following equivalent statements I have found:
- the column vectors without the first entry are affinely independent
- the convex hull of $A$ without the first row has the full dimension $n$
And I tried (currently without success) to interprete $A$ as an incidence matrix of a hypergraph with $K$- and $(K+1)$-regular part.
Thanks a lot
No. $$ \pmatrix{ 1&1&1&1&1\\ 0&1&1&1&0\\ 1&0&1&1&1\\ 1&1&0&0&1\\ 1&1&1&0&0}\pmatrix{1\\ 0\\ -1\\ 1\\ -1}=0. $$