Deconvolution (Linear Convolution) with an Under Determined System of Equations?
On the contrary, if you compute the convolution of a signal $x$ having a span of $M$ non-zero entries with a filter with a span of $N$ non-zero entries, then the resulting sequence will have $M+N-1$ non-zero entries, so that your system should be over-determined.

Computing the minimizer $x$ of $\|[0,1,0]*y-[1,1,1]*x\|_2$ leads by standard variational methods to the system of equations

$$[0,1,1,1,0]*y=[1,2,3,2,1]*x$$

with the first two and the last two removed, and for some reason mathematics / linear algebra claims that this system is solvable even if $[0,1,0]*y=[1,1,1]*x$ is not.