I was toying around with an old problem, how to generalize the fundamental theorem of arithmetic to multivariate polynomials.
A very simplex example of 2-variate polynomial that cannot be factored into a product of planes of the form $a_0 + a_1 x + a_2y$ is
$$ y^2 - x$$
If one attempts to actually find $(a_0 + a_1 x + a_2y)(b_0 + b_1 x + b_2y)$ to be equal to that ie.
$$ a_0 b_0 = 0$$ $$ a_0 b_1 + a_1 b_0 = -1$$ $$ a_0 b_2 + a_2 b_0 = 0$$ $$ a_1 b_1 = 0$$ $$a_1 b_2 + a_2 b_1 = 0$$ $$ a_2 b_2 = 1$$
One finds that the system is unsolvable, and the crux of it really comes down to the fact that if $ab = 0$ over $\mathbb{C}$ then one of $a$ or $b$ is 0.
Now that had me curious, if one looks at a higher level, say to the space of $\mathbb{C}^{2\times2}$ matrices, could it be possible to wrangle a solution out of this? (We fix the order written above, since matrix multiplication isn't commutative)
I would've done this in an automated way to check, but unfortunately doing it coefficient by coefficient leads to a system of 24 quadratic equations in 24 variables which will definitely exceed my RAM (solution space has $2^{24}$ solutions, if it is solvable, counting multiplicity)
So at this point, I'm forced to ask the theoretical question before I get any experimental evidence.
From $a_2 b_2 = 1$ we get $b_2 = a_2^{-1}$. Then from $a_0 b_2 + a_2 b_0 = 0$ we get $a_0 = - a_2 b_0 b_2^{-1} = - a_2 b_0 a_2$, and from $a_1 b_2 + a_2 b_1 = 0$ we get $a_1 = - a_2 b_1 b_2^{-1} = -a_2 b_1 a_2$. Now $0 = a_0 b_0 = - a_2 b_0 a_2 b_0$, but $a_2$ is invertible so this says $b_0 a_2 b_0 = 0$. Similarly $0 = a_1 b_1 = - a_2 b_1 a_2 b_1$, so $b_1 a_2 b_1 = 0$. The remaining equation is $ a_0 b_1 + a_1 b_0 = - a_2 b_0 a_2 b_1 - a_2 b_1 a_2 b_0 = -1$. Maple finds a number of solution families in $2 \times 2$ complex matrices, e.g., if I haven't made a mistake, $$ a_0 = \pmatrix{-tu/(s^2 v) & t/(sv)\cr u/(sv) & -1/v\cr},\ a_1 = \pmatrix{-1/s & 0\cr 0 & 0\cr},\ a_2 = \pmatrix{-(t+r^2suv)/(rs^2 v) & r\cr 1/(rsv) & 0\cr},$$ $$b_0 = \pmatrix{s & t\cr u & tu/s\cr}\ b_1 = \pmatrix{0 & 0\cr 0 & v\cr}\ b_2 = \pmatrix{0 & rsv\cr 1/r & (r^2suv+t)/(rs)\cr}$$