For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w\_1,x\_1,y\_1,z\_1,w\_2,x\_2,y\_2,z\_2$ with absolute value less than $T$ such that
$$w\_1x\_1+y\_1z\_1=a$$
$$w\_2x\_1+y\_2z\_1=b$$
$$w\_1x\_2+y\_1z\_2=c$$
$$w\_2x\_2+y\_2z\_2=d$$
holds?


If not for what fraction of $a,b,c,d$ does it fail?


Typically how many solutions do we get?



Representing integers efficiently with quadratic polynomials
Here is an answer. If it inspires you to change the question, please don't. Just ask a new one.


No, you can't even say that the $8$ unknowns are bounded by $T^2-1.$ 


To recast, given an integer matrix $A=\begin{pmatrix} 
a & c \\
b & d 
\end{pmatrix}$ with entries bounded in absolute value by $T^2,$ you wonder about finding integer matrices $W=\begin{pmatrix} 
w\_1 & y\_1 \\
w\_2 & y\_2 
\end{pmatrix}$ and $X=\begin{pmatrix} 
x\_1 & x\_2 \\
z\_1 & z\_2 
\end{pmatrix}$ with smaller entries and $WX=A.$


Since $\det(W)\det(X)=\det(A)$ there are constraints. For example with $T=1000$ and $p=999983$ consider 


$$\begin{pmatrix} 
w\_1 & y\_1 \\
w\_2 & y\_2 
\end{pmatrix}\begin{pmatrix} 
x\_1 & x\_2 \\
z\_1 & z\_2 
\end{pmatrix}=\begin{pmatrix} 
p & 0 \\
0 & 1 
\end{pmatrix}.$$


As $p$ is prime, one of the two factors has determinant $1$, say the second. Then you can work out that 


$$\begin{pmatrix} 
w\_1 & y\_1 \\
w\_2 & y\_2 
\end{pmatrix}=\begin{pmatrix} 
p & 0 \\
0 & 1 
\end{pmatrix}\begin{pmatrix} 
z\_2 & -x\_2 \\
-z\_1 & x\_1 
\end{pmatrix}=\begin{pmatrix} 
pz\_2 & -px\_2 \\
z\_1 & x\_1 
\end{pmatrix}.$$


———


I wonder what a counting argument would give. There are about $16T^8$ ways to pick the $4$ parameters and about $256T^8$ ways to pick the $8$ unknowns. So a $16:1$ ratio. But that drops to $2:1$ once you factor in that you can swap the roles of the various $u\_1,u\_2$ and/or negate some pairs.


In many cases there are enormously more solutions . This should mean that other times there are none. 


Also, frequently the product of two matrices with entries bounded by $T$ will contain entries larger than $T^2$ (though no larger than $2T^2.$)


