I am working on a problem involving nilpotent matrices over $\mathbb{F}\_2$ and I was able to reduce it to proving that the system
\begin{equation}
 \begin{cases}
 A^2+ BC+ BCA+ ABC+A = I\_4 \\
 AB+ABD+BCB = 0 \\
 CA+DCA+CBC = 0 \\
 DCB+CBD = I\_4 \\
 A^3+BCA+ABC+BDC=0 \\
 A^2B+BCB+ABD+BD^2=0 \\
 CA^2+DCA+CBC+D^2C=0 \\
 CAB+DCB+CBD+D^3=0
 \end{cases}\,,
\end{equation}
has no solution, where $A, B, C, D$ are $4 \times 4$ matrices over $\mathbb{F}\_2.$
The first four equations were obtained by plugging-in matrices to some polynomial in $\mathbb{F}\_2$ while the other four came from the condition that
$$\left[
\begin{array}{c|c}
A & B \\
\hline
C & D
\end{array}
\right]^3 = 0.$$


Any advice on how to prove this? Any help would be appreciated.


The original problem is if
$$M=\begin{bmatrix} N & 0 \\ 0 & N \end{bmatrix}, \mbox{ where } N=\begin{bmatrix} 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1\end{bmatrix}$$
then $M$ cannot be written as a sum $X+Y$ such that $X^3=0$ and $Y^2=Y.$ Here $M, X, Y$ are matrices over $\mathbb{F}\_2$.


Write $X=[x\_{i,j}], Y=[y\_{i,j}]$.
For each $(i,j)$, $i,j \in \{1, \ldots, 8\}$, $x\_{i,j}$ and $y\_{i,j}$ satisfies the following equations:
\begin{equation}
 \begin{cases}
 \displaystyle \sum\_{k=1}^8\sum\_{l=1}^8 x\_{i,k}x\_{k,l}x\_{l,j}=0\\
 \displaystyle \sum\_{k=1}^8 y\_{i,k}y\_{k,j}=y\_{i,j}\\ 
 x\_{i,j}+y\_{i,j}=m\_{i,j} \\
 x\_{i,j}^2= x\_{i,j} \\
 y\_{i,j}^2= y\_{i,j}
 \end{cases}\,,
\end{equation}


So we want to show that this system has no solution via Groebner basis and implement it using Maple. I have zero background in Maple so I need help with the code. So what I've researched is we start with



```
with(Groebner);

```

and define matrix $M$:



```
M:=[[0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0], [0,0,1,1,0,0,0,0], [0,0,0,0,0,0,0,1], [0,0,0,0,1,0,0,0], [0,0,0,0,0,1,0,0], [0,0,0,0,0,0,1,1]];

```

Define the following polynomials for all $(i,j)$:



```
f_{i,j}= \sum_{k=1}^8\sum_{l=1}^8 x_{i,k}x_{k,l}x_{l,j};
g_{i,j}= \sum_{k=1}^8 y_{i,k}y_{k,j}-y_{i,j};
h_{i,j}=x_{i,j}+y_{i,j}-M[i,j];
q_{i,j}=x_{i,j}^2-x_{i,j};
s_{i,j}=y_{i,j}^2-y_{i,j};

```

and we let for all $(i,j)$



```
B:=[f_{i,j},g_{i,j},h_{i,j},q_{i,j},s_{i,j}];

```

and call



```
G:=Groebner[Basis](B, plex(x_{i,j},y_{i,j}), characteristic=2);

```

We expect that the output is



```
G:=[1]

```

Is this correct? Also, how do I typeset the summations in Maple, and how do I typeset the polynomials so that I don't have to manually type for all 64 pairs $(i,j)$? Thank you so much.



Proving that a system of polynomial matrix equations over $\mathbb{F_2}$ has no solution
We can solve quickly this problem using the basis Grobner theory.


Put $X=[x\_{i,j}],Y=[y\_{i,j}]$.


We consider -over a field of characteristic $2$- the algebraic system in the $128$ unknowns $x\_{i,j},y\_{i,j}$ constituted by the equations $M=X+Y,X^3=0,Y^2=Y$ -entrywise- and $x\_{i,j}^2=x\_{i,j},y\_{i,j}^2=y\_{i,j}$.


With the help of the FGb library of Maple, we obtain (in 36") that there are no solutions.


$\textbf{Answer to the OP.}$ The maple command "with(Groebner)" is not very efficient; prefer the command "with(FGb)". You need to load the patch here (available only on LINUX)


<http://www.mathemagix.org/www/mfgb/doc/html/install_fgb.en.html>


Otherwise use "with(Groebner) but it may be a long way.


You can also use "Sage" very powerful but not very practical to use.
Here is the program I used.


restart:


with(LinearAlgebra):


n := 8:


N := Matrix([[0, 0, 0, 1], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1]]):


M := DiagonalMatrix([N, N]):


X := Matrix(8, symbol = x):


Y := Matrix(8, symbol = y):


C := M-X-Y: E := X^3: P := Y^2-Y:


K := NULL:


for i to n do


for j to n do


K := K, x[i, j], y[i, j] end do end do:


K := [K]:


F := NULL:


for i to n do


for j to n do


F := F, C[i, j], E[i, j], P[i, j], x[i, j]^2-x[i, j], y[i, j]^2-y[i, j] end do end do:


F := [F]:


with(FGb):


t := time():


solu := fgb\_gbasis(F, 2, K, [], {"index" = 10^7, "verb" = 3});


nops(solu):


t3 := time()-t:


